LaTeX vs MathPix image — 0902.0431

\tau(u+i v)=u-i v .

\tau(u + iv) = u - iv.

\gamma, \quad \sigma, \quad \iota, \quad \kappa, \quad \lambda, \quad \mu, \quad \tau, \quad v, \quad \chi, \quad g, \quad w .

\gamma, \quad \sigma, \quad \iota, \quad \kappa, \quad \lambda, \quad \mu, \quad \tau, \quad \upsilon, \quad \chi, \quad g, \quad w.

e_{1} e_{2}=e_{3}, \quad e_{2} e_{3}=e_{1}, \quad e_{3} e_{1}=e_{2},

e_1e_2 = e_3, \quad e_2e_3 = e_1, \quad e_3e_1 = e_2,

e_{i}^{2}=-1, \quad i \neq 0, \quad e_{i} e_{j}=-e_{j} e_{i}, \quad i \neq j, i \neq 0, j \neq 0,

{e_i}^2 = -1 ,\;\; i \neq 0, \quad e_ie_j = -e_je_i ,\;\; i \neq j , i \neq 0 , j \neq 0,

\begin{array}{cr} \overline{x_{0}+\sum_{i=1}^{7} x_{i} e_{i}}=x_{0}-\sum_{i=1}^{7} x_{i} e_{i}, & \left(\sum_{i=0}^{7} x_{i} e_{i}, \sum_{i=0}^{7} y_{i} e_{i}\right)=\sum_{i=0}^{7} x_{i} y_{i} \\ |x|=\sqrt{(x, x)}, & R\left(x_{0}+\sum_{i=1}^{7} x_{i} e_{i}\right)=x_{0} . \end{array}

\begin{array}{cc} \overline{x_0 + \sum_{i=1}^7x_ie_i} = x_0 - \sum_{i=1}^7x_ie_i, & \Big(\sum_{i=0}^7x_ie_i, \sum_{i=0}^7y_ie_i \Big) = \sum_{i=0}^7x_iy_i, \vspace{1mm}\\ |x| = \sqrt{(x, x)}, & R\Big(x_0 + \sum_{i=1}^7x_ie_i \Big) = x_0. \end{array}

\{x, y, a\}=\{y, a, x\}=\{a, x, y\}=-\{y, x, a\}=-\{x, a, y\}=-\{a, y, x\} .

\begin{aligned} & (a x) y+x(y a)=a(x y)+(x y) a, \\ & (x a) y+(x y) a=x(a y)+x(y a), \\ & (a x) y+(x a) y=a(x y)+x(a y) . \end{aligned}

\begin{array}{l} (ax)y + x(ya) = a(xy) + (xy)a, \vspace{1mm}\\ (xa)y + (xy)a = x(ay) + x(ya), \vspace{1mm}\\ (ax)y + (xa)y = a(xy) + x(ay). \end{array}

G_{2}=\left\{\alpha \in \operatorname{Iso}_{\boldsymbol{R}}(\mathfrak{C}) \mid \alpha(x y)=(\alpha x)(\alpha y)\right\} .

G_2 = \{ \alpha \in \text\mathrm{{Iso}}_{\text{${R}$}}(\text{\es {C}}) \, | \, \alpha(xy) = (\alpha x)(\alpha y) \}.

(\alpha x, \alpha y)=(x, y), \quad x, y \in \mathfrak{C} .

(\alpha x, \alpha y) = (x,y), \quad x,y \in \text{\es {C}}.

\overline{\alpha x}=\alpha \bar{x} .

\overline{\alpha x} = \alpha\overline{x}.

\alpha 1=1, \quad \overline{\alpha e_{i}}=-\alpha e_{i}, \quad i=1,2, \cdots, 7 .

\alpha 1 = 1,\;\; \overline{\alpha e_i} = - \alpha e_i,\quad i=1,2,\cdots,7.

\begin{aligned} (\alpha x, \alpha y) & =\frac{1}{2}((\alpha x)(\overline{\alpha y})+(\alpha y)(\overline{\alpha x}))=\frac{1}{2}((\alpha x)(\alpha \bar{y})+(\alpha y)(\alpha \bar{x})) \\ & =\alpha\left(\frac{1}{2}(x \bar{y}+y \bar{x})\right)=\alpha((x, y))=(x, y) . \end{aligned}

\begin{aligned} (\alpha x,\alpha y) \!\!\! &=& \!\!\! \frac{1}{2}((\alpha x)(\overline{\alpha y}) + (\alpha y)(\overline{\alpha x})) = \frac{1}{2}((\alpha x)(\alpha \overline{y}) + (\alpha y)(\alpha \overline{x})) \vspace{1mm}\\ \!\!\! &=& \!\!\! \alpha\Big(\frac{1}{2}(x\overline{y} + y\overline{x})\Big) = \alpha((x,y)) = (x,y). \end{aligned}

O(8)=O(\mathfrak{C})=\left\{\alpha \in \operatorname{Iso}_{\boldsymbol{R}}(\mathfrak{C}) \mid(\alpha x, \alpha y)=(x, y)\right\} .

O(8) = O(\text{\es {C}}) = \{ \alpha \in \text\mathrm{{Iso}}_{\text{${R}$}}(\text{\es {C}}) \, | \, (\alpha x, \alpha y) =(x, y) \}.

\mathfrak{D}_{4}=\mathfrak{s o}(8)=\mathfrak{s o}(\mathfrak{C})=\left\{D \in \operatorname{Hom}_{\boldsymbol{R}}(\mathfrak{C}) \mid(D x, y)+(x, D y)=0\right\}

\text{\es {D}}_4 = \text{\es {so}}(8) = \text{\es {so}}(\text{\es {C}}) = \{ D \in \text\mathrm{{Hom}}_{\text{${R}$}}(\text{\es {C}}) \, | \, (Dx, y) + (x, Dy) = 0 \}

G_{i j} e_{j}=e_{i}, \quad G_{i j} e_{i}=-e_{j}, \quad G_{i j} e_{k}=0, k \neq i, j .

G_{ij}e_j = e_i,\;\;\; G_{ij}e_i = -e_j,\;\;\; G_{ij}e_k = 0, \;\, k \neq i,j.

F_{i j} x=\frac{1}{2} e_{i}\left(\bar{e}_{j} x\right), \quad x \in \mathfrak{C} .

F_{ij}x = {\frac{1}{2}}e_i(\overline{e}_jx), \quad x \in \text{\es {C}}.

\left\{\begin{array} { l } { 2 F _ { 0 1 } = G _ { 0 1 } + G _ { 2 3 } + G _ { 4 5 } + G _ { 6 7 } } \\ { 2 F _ { 2 3 } = G _ { 0 1 } + G _ { 2 3 } - G _ { 4 5 } - G _ { 6 7 } } \\ { 2 F _ { 4 5 } = G _ { 0 1 } - G _ { 2 3 } + G _ { 4 5 } - G _ { 6 7 } } \\ { 2 F _ { 6 7 } = G _ { 0 1 } - G _ { 2 3 } - G _ { 4 5 } + G _ { 6 7 } , } \end{array} \left\{\begin{array}{l} 2 F_{02}=G_{02}-G_{13}-G_{46}+G_{57} \\ 2 F_{13}=-G_{02}+G_{13}-G_{46}+G_{57} \\ 2 F_{46}=-G_{02}-G_{13}+G_{46}+G_{57} \\ 2 F_{57}=G_{02}+G_{13}+G_{46}+G_{57}, \end{array}\right.\right.

\begin{gathered} \left\{\begin{array} { l } { 2 F _ { 0 3 } = G _ { 0 3 } + G _ { 1 2 } + G _ { 4 7 } + G _ { 5 6 } } \\ { 2 F _ { 1 2 } = G _ { 0 3 } + G _ { 1 2 } - G _ { 4 7 } - G _ { 5 6 } } \\ { 2 F _ { 4 7 } = G _ { 0 3 } - G _ { 1 2 } + G _ { 4 7 } - G _ { 5 6 } } \\ { 2 F _ { 5 6 } = G _ { 0 3 } - G _ { 1 2 } - G _ { 4 7 } + G _ { 5 6 } , } \end{array} \left\{\begin{array}{l} 2 F_{04}=G_{04}-G_{15}+G_{26}-G_{37} \\ 2 F_{15}=-G_{04}+G_{15}+G_{26}-G_{37} \\ 2 F_{26}=G_{04}+G_{15}+G_{26}+G_{37} \\ 2 F_{37}=-G_{04}-G_{15}+G_{26}+G_{37}, \end{array}\right.\right. \\ \left\{\begin{array} { l } { 2 F _ { 0 5 } = G _ { 0 5 } + G _ { 1 4 } - G _ { 2 7 } - G _ { 3 6 } } \\ { 2 F _ { 1 4 } = G _ { 0 5 } + G _ { 1 4 } + G _ { 2 7 } + G _ { 3 6 } } \\ { 2 F _ { 2 7 } = - G _ { 0 5 } + G _ { 1 4 } + G _ { 2 7 } - G _ { 3 6 } } \\ { 2 F _ { 3 6 } = - G _ { 0 5 } + G _ { 1 4 } - G _ { 2 7 } + G _ { 3 6 } , } \end{array} \left\{\begin{array}{l} 2 F_{06}=G_{06}-G_{17}-G_{24}+G_{35} \\ 2 F_{17}=-G_{06}+G_{17}-G_{24}+G_{35} \\ 2 F_{24}=-G_{06}-G_{17}+G_{24}+G_{35} \\ 2 F_{35}=G_{06}+G_{17}+G_{24}+G_{35}, \end{array}\right.\right. \\ \left\{\begin{array}{l} 2 F_{07}=G_{07}+G_{16}+G_{25}+G_{34} \\ 2 F_{16}=G_{07}+G_{16}-G_{25}-G_{34} \\ 2 F_{25}=G_{07}-G_{16}+G_{25}-G_{34} \\ 2 F_{34}=G_{07}-G_{16}-G_{25}+G_{34} . \end{array}\right. \end{gathered}

\begin{aligned} (\kappa D) x & =\overline{D \bar{x}}, \quad x \in \mathfrak{C}, \\ \pi\left(G_{i j}\right) & =F_{i j}, \quad i, j=0,1, \cdots, 7, i \neq j, \\ \nu & =\pi \kappa . \end{aligned}

\begin{aligned} (\kappa D)x \!\! &=& \!\! \overline{D\overline{x}}, \quad x \in \text{\es {C}}, \\ \pi(G_{ij}) \!\! &=& \!\! F_{ij},\quad i,j = 0,1,\cdots, 7, i \neq j, \\ \nu \!\! &=& \!\! \pi\kappa. \end{aligned}

\kappa, \pi, \nu \in \operatorname{Aut}\left(\mathfrak{D}_{4}\right) .

\kappa,\pi,\nu \in \text{Aut}(\text{\es {D}}_4).

\begin{aligned} & {\left[\kappa D_{1}, \kappa D_{2}\right] x=\left(\kappa D_{1}\right)\left(\kappa D_{2} x\right)-\left(\kappa D_{2}\right)\left(\kappa D_{1} x\right)=\overline{D_{1}\left(\overline{\kappa D_{2} x}\right)}-\overline{D_{2}\left(\overline{\kappa D_{1} x}\right)}} \\ & \quad=\overline{D_{1} D_{2} \bar{x}}-\overline{D_{2} D_{1} \bar{x}}=\kappa\left(D_{1} D_{2}-D_{2} D_{1}\right) x=\kappa\left[D_{1}, D_{2}\right] x, \quad x \in \mathfrak{C} \end{aligned}

\begin{array}{l} [\kappa D_1,\kappa D_2]x = (\kappa D_1)(\kappa D_2x) - (\kappa D_2)(\kappa D_1x) = \overline{D_1(\overline{\kappa D_2x})} - \overline{D_2(\overline{\kappa D_1 x})} \vspace{1mm} \\ \qquad \quad = \overline{D_1D_2\overline{x}}-\overline{D_2D_1\overline{x}} = \kappa(D_1D_2 - D_2D_1)x = \kappa[D_1,D_2]x,\quad x \in \text{\es {C}}. \end{array}

\left[\pi G_{i j}, \pi G_{k l}\right]=\pi\left[G_{i j}, G_{k l}\right],

[\pi G_{ij}, \pi G_{kl}] = \pi[G_{ij}, G_{kl}],

\begin{array}{ll} {\left[F_{i j}, F_{j k}\right]=F_{i k},} & i, j, k \text { are distinct } \\ {\left[F_{i j}, F_{k l}\right]=0,} & i, j, k, l \text { are distinct. } \end{array}

\begin{array}{ll} [F_{ij}, F_{jk}] = F_{ik}, & \quad i,j,k \; \text{are distinct}, \vspace{1mm}\\ {[}F_{ij}, F_{kl}{]} = 0, & \quad i,j,k,l \; \text{are distinct}. \end{array}

\begin{aligned} {\left[F_{i 0}, F_{0 k}\right] x } & =\left(F_{i 0} F_{0 k}-F_{0 k} F_{i 0}\right) x=-\frac{1}{4} e_{i}\left(e_{k} x\right)+\frac{1}{4} e_{k}\left(e_{i} x\right) \\ & =\frac{1}{2} e_{k}\left(e_{i} x\right)=F_{i k} x, \end{aligned}

\begin{array}{l} [F_{i0}, F_{0k}]x = (F_{i0}F_{0k}- F_{0k}F_{i0})x = - {\frac{1} {4}}e_i(e_k x) + \frac{1}{4}e_k(e_i x) \vspace{1mm}\\ \qquad \qquad \;\;\; = \frac{1}{2}e_k (e_i x) = F_{ik}x, \end{array}

\begin{aligned} {\left[F_{i 0}, F_{k l}\right] x } & =\left(F_{i 0} F_{k l}-F_{k l} F_{i 0}\right) x=\frac{1}{4} e_{i}\left(e_{l}\left(e_{k} x\right)\right)-\frac{1}{4} e_{l}\left(e_{k}\left(e_{i} x\right)\right) \\ & =-\frac{1}{4} e_{l}\left(e_{i}\left(e_{k} x\right)\right)+\frac{1}{4} e_{l}\left(e_{i}\left(e_{k} x\right)\right)=0, \\ {\left[F_{i j}, F_{j k}\right] x } & =\left(F_{i j} F_{j k}-F_{j k} F_{i j}\right) x=\frac{1}{4} e_{j}\left(e_{i}\left(e_{k}\left(e_{j} x\right)\right)\right)-\frac{1}{4} e_{k}\left(e_{j}\left(e_{j}\left(e_{i} x\right)\right)\right) \\ & =\frac{1}{4} e_{i}\left(e_{k}\left(e_{j}\left(e_{j} x\right)\right)\right)+\frac{1}{4} e_{k}\left(e_{i} x\right)=-\frac{1}{4} e_{i}\left(e_{k} x\right)+\frac{1}{4} e_{k}\left(e_{i} x\right) \\ & =\frac{1}{2} e_{k}\left(e_{i} x\right)=F_{i k} x, \\ {\left[F_{i j}, F_{k l}\right] x } & =\left(F_{i j} F_{k l}-F_{k l} F_{i j}\right) x=\frac{1}{4} e_{j}\left(e_{i}\left(e_{l}\left(e_{k} x\right)\right)\right)-\frac{1}{4} e_{l}\left(e_{k}\left(e_{j}\left(e_{i} x\right)\right)\right) \\ & =\frac{1}{4} e_{j}\left(e_{k}\left(e_{i}\left(e_{l} x\right)\right)\right)-\frac{1}{4} e_{j}\left(e_{l}\left(e_{k}\left(e_{i} x\right)\right)\right)=\cdots=0 . \end{aligned}

\begin{aligned} {[}F_{i0}, F_{kl}{]}x \!\!\! &=& \!\!\! (F_{i0}F_{kl} - F_{kl}F_{i0})x = \frac{1}{4}e_i (e_l (e_k x)) - \frac{1}{4}e_l(e_k(e_i x))\\ \!\!\! &=& \!\!\! -\frac{1}{4}e_l (e_i(e_k x)) + \frac{1}{4}e_l(e_i(e_k x)) = 0,\\ {[}F_{ij}, F_{jk}{]}x \!\!\! &=& \!\!\! (F_{ij}F_{jk} - F_{jk}F_{ij})x = \frac{1}{4}e_j (e_i(e_k(e_jx))) - \frac{1}{4}e_k(e_j(e_j(e_ix)))\\ \!\!\! &=& \!\!\! \frac{1}{4}e_i(e_k(e_j(e_jx))) + \frac{1}{4}e_k(e_ix) = -\frac{1}{4}e_i(e_kx) + \frac{1}{4}e_k(e_ix)\\ \!\!\! &=& \!\!\! \frac{1}{2}e_k(e_ix) = F_{ik}x,\\ {[}F_{ij}, F_{kl}{]}x \!\!\! &=& \!\!\! (F_{ij}F_{kl} - F_{kl}F_{ij})x = \frac{1}{4}e_j (e_i(e_l(e_kx))) - \frac{1}{4}e_l(e_k(e_j(e_ix)))\\ \!\!\! &=& \!\!\! \frac{1}{4}e_j(e_k(e_i(e_lx))) - \frac{1}{4}e_j(e_l(e_k(e_ix)))= \cdots = 0. \end{aligned}

L_{a} x=a x, \quad R_{a} x=x a, \quad T_{a} x=a x+x a=\left(L_{a}+R_{a}\right) x, \quad x \in \mathfrak{C} .

L_ax = ax , \quad R_ax = xa , \quad T_ax = ax + xa = (L_a + R_a)x, \quad x \in \text{\es {C}}.

L_{a}, R_{a}, T_{a} \in \mathfrak{D}_{4} .

\kappa L_{a}=-R_{a}, \quad \kappa R_{a}=-L_{a}, \quad \kappa T_{a}=-T_{a} .

\pi L_{a}=T_{a}, \quad \pi R_{a}=-R_{a}, \quad \pi T_{a}=L_{a} .

\nu L_{a}=R_{a}, \quad \nu R_{a}=-T_{a}, \quad \nu T_{a}=-L_{a} .

L_{e_{i}}=2 F_{i 0}, \quad T_{e_{i}}=2 G_{i 0} .

L_{e_i} = 2F_{i0}, \quad T_{e_i} = 2G_{i0}.

\begin{aligned} & L_{e_{i}} x=e_{i} x=2 F_{i 0} x, \quad x \in \mathfrak{C}, \\ & T_{e_{i}} x=e_{i} x+x e_{i}= \begin{cases}2 e_{i}, & x=1 \\ -2, & x=e_{i} \\ 0, & x=e_{j}, \quad j \neq 0, i .\end{cases} \end{aligned}

\begin{array}{l} L_{e_i}x = e_ix = 2F_{i0}x, \quad x \in \text{\es {C}}, \vspace{1mm}\\ T_{e_i}x = e_ix + xe_i = \left\{\begin{array}{ll} 2e_i, & \quad x = 1 \\ -2, & \quad x = e_i \\ 0, & \quad x = e_j,\;\; j \neq 0,i. \end{array}\right. \end{array}

\pi L_{e_{i}}=\pi\left(2 F_{i 0}\right)=-2 \pi F_{0 i}=-2 G_{0 i}(\text { Lemma 1.3.1 })=2 G_{i 0}=T_{e_{i}},

\begin{aligned} & \pi T_{e_{i}}=\pi\left(2 G_{i 0}\right)=2 F_{i 0}=L_{e_{i}}, \\ & \pi R_{a}=\pi\left(T_{a}-L_{a}\right)=L_{a}-T_{a}=-R_{a} . \end{aligned}

D=L_{a}+\sum_{i}\left[L_{b_{i}}, L_{c_{i}}\right], \quad a, b_{i}, c_{i} \in \mathfrak{C}_{0} .

D = L_a + \sum_{i}[L_{b_i}, L_{c_i}], \quad a,b_i,c_i \in \text{\es {C}}_0.

\kappa^{2}=1, \quad \pi^{2}=1, \quad \nu^{3}=1, \quad \nu=\pi \kappa .

\kappa^2 = 1, \quad \pi^2 = 1, \quad \nu^3 = 1, \quad \nu =\pi \kappa.

\begin{array}{ll} 1 \rightarrow\left(\begin{array}{lll} 1 & 2 & 3 \\ 1 & 2 & 3 \end{array}\right), \quad \kappa \rightarrow\left(\begin{array}{lll} 1 & 2 & 3 \\ 2 & 1 & 3 \end{array}\right), \quad \pi \rightarrow\left(\begin{array}{lll} 1 & 2 & 3 \\ 3 & 2 & 1 \end{array}\right), \\ \nu \rightarrow\left(\begin{array}{lll} 1 & 2 & 3 \\ 2 & 3 & 1 \end{array}\right), \quad \nu^{2} \rightarrow\left(\begin{array}{lll} 1 & 2 & 3 \\ 3 & 1 & 2 \end{array}\right), \quad \nu \pi \rightarrow\left(\begin{array}{lll} 1 & 2 & 3 \\ 1 & 3 & 2 \end{array}\right) \end{array}

\left(D_{1} x\right) y+x\left(D_{2} y\right)=D_{3}(x y), \quad x, y \in \mathfrak{C} .

(D_1x)y + x(D_2y) = D_3(xy), \quad x,y \in \text{\es {C}}.

D_{2}=\nu D_{1}, \quad D_{3}=\pi D_{1} .

D_2 = \nu D_1, \quad D_3 = \pi D_1.

\left(L_{a} x\right) y+x\left(R_{a} y\right)=T_{a}(x y), \quad x, y \in \mathfrak{C} .

\displaylines{\hfill (L_ax)y + x(R_ay) = T_a(xy), \quad x,y \in \text{\es {C}}. \hfill\text{(i)}}

\left(L_{b} x\right) y+x\left(R_{b} y\right)=T_{b}(x y), \quad x, y \in \mathfrak{C} .

\displaylines{\hfill (L_bx)y + x(R_by) = T_b(xy), \quad x,y \in \text{\es {C}}. \hfill\text{(ii)}}

\left(L_{a} L_{b} x\right) y+\left(L_{b} x\right)\left(R_{a} y\right)+\left(L_{a} x\right)\left(R_{b} y\right)+x\left(R_{a} R_{b} y\right)=T_{a} T_{b}(x y) .

(L_aL_bx)y + (L_bx)(R_ay) + (L_ax)(R_by) + x(R_aR_by) =T _aT_b(xy).

\left(\left[L_{a}, L_{b}\right] x\right) y+x\left(\left[R_{a}, R_{b}\right] y\right)=\left[T_{a}, T_{b}\right](x y) .

\displaylines{\hfill ([L_a,L_b]x)y + x([R_a,R_b]y) = [T_a,T_b](xy). \hfill\text{(iii)}}

D_{1}=L_{a}+\sum\left[L_{b}, L_{c}\right], \quad a, b, c \in \mathfrak{C}_{0}

D_1 = L_a + \sum[L_b,L_c], \quad a,b,c \in \text{\es {C}}_0

\left(D_{1} x\right) y+x\left(D_{2} y\right)=D_{3}(x y), \quad x, y \in \mathfrak{C} .

(D_1x)y + x(D_2y) = D_3(xy), \quad x,y \in \text{\es {C}}.

x\left(D_{2} y\right)=D_{3}(x y), \quad x, y \in \mathfrak{C} .

x(D_2y) = D_3(xy), \quad x,y \in \text{\es {C}}.

x(D y)=D(x y), \quad x, y \in \mathfrak{C} .

\displaylines{\hfill x(Dy) = D(xy), \quad x,y \in \text{\es {C}}. \hfill\text{(iv)}}

x(y p)=(x y) p, \quad \text { for all } x, y \in \mathfrak{C} .

x(yp)=(xy)p,\quad \text{for all}\;\; x,y \in \text{\es {C}}.

\left(D_{1} x\right) y+x\left(D_{2} y\right)=\left(\kappa D_{3}\right)(x y), \quad x, y \in \mathfrak{C}

(D_1x)y + x(D_2y) = (\kappa D_3)(xy), \quad x,y \in \text{\es {C}}

\begin{array}{ll} \left(D_{2} x\right) y+x\left(D_{3} y\right)=\left(\kappa D_{1}\right)(x y), & x, y \in \mathfrak{C}, \\ \left(D_{3} x\right) y+x\left(D_{1} y\right)=\left(\kappa D_{2}\right)(x y), & x, y \in \mathfrak{C} . \end{array}

\begin{array}{l} (D_2x)y + x(D_3y) = (\kappa D_1)(xy), \quad x,y \in \text{\es {C}}, \vspace{1mm}\\ (D_3x)y + x(D_1y) = (\kappa D_2)(xy), \quad x,y \in \text{\es {C}}. \end{array}

\left(D_{2} x\right) y+x\left(D_{3}^{\prime} y\right)=\left(\kappa D_{1}^{\prime}\right)(x y), \quad x, y \in \mathfrak{C},

(D_2x)y + x({D_3}'y) = (\kappa {D_1}')(xy), \quad x,y \in \text{\es {C}},

\begin{aligned} & D_{3}^{\prime}=\nu D_{2}=\nu \nu D_{1}=\nu^{-1} D_{1}=\kappa \pi D_{1}=\kappa \kappa D_{3}=D_{3} \\ & D_{1}^{\prime}=\kappa \pi D_{2}=\nu^{-1} D_{2}=D_{1} \end{aligned}

\begin{array}{l} {D_3}' = \nu D_2 = \nu\nu D_1 = \nu^{-1}D_1 = \kappa\pi D_1 = \kappa\kappa D_3 = D_3, \vspace{1mm} \\ {D_1}' = \kappa\pi D_2 = \nu^{-1}D_2 = D_1. \end{array}

\mathfrak{g}_{2}=\left\{D \in \operatorname{Hom}_{\boldsymbol{R}}(\mathfrak{C}) \mid D(x y)=(D x) y+x(D y)\right\} .

\text{\es {g}}_2 = \{ D \in \text\mathrm{{Hom}}_{\text{${R}$}}(\text{\es {C}}) \, | \, D(xy) = (Dx)y + x(Dy) \}.

\begin{aligned} \mathfrak{b}_{3} & =\left\{D \in \mathfrak{D}_{4} \mid D 1=0\right\} \\ & =\left\{D \in \mathfrak{D}_{4} \mid \kappa D=D\right\}=\left\{D \in \mathfrak{D}_{4} \mid \nu D=\pi D\right\} . \end{aligned}

\begin{array}{lll} \text{\es {b}}_3 \!\!\! &=& \!\!\! \{ D \in \text{\es {D}}_4 \, | \, D1 = 0 \} \vspace{1mm}\\ \!\!\! &=& \!\!\! \{ D \in \text{\es {D}}_4 \, | \, \kappa D = D \} = \{ D \in \text{\es {D}}_4\, | \, \nu D = \pi D \}. \vspace{3mm} \end{array}

\begin{aligned} \mathfrak{g}_{2} & =\left\{D \in \mathfrak{D}_{4} \mid \nu D=D, \pi D=D\right\} \\ & =\left\{D \in \mathfrak{D}_{4} \mid \lambda D=D, \lambda \in \mathfrak{S}_{3}\right\} \\ & =\left\{D \in \mathfrak{b}_{3} \mid \pi D=D\right\} . \end{aligned}

\begin{aligned} \text{\es {g}}_2 \!\!\! &=& \!\!\! \{ D \in \text{\es {D}}_4 \, | \, \nu D = D, \pi D = D \} \\ \!\!\! &=& \!\!\! \{ D \in \text{\es {D}}_4 \, | \, \lambda D = D, \lambda \in \text{\es {S}}_3 \} \\ \!\!\! &=& \!\!\! \{ D \in \text{\es {b}}_3 \, | \, \pi D = D \}. \end{aligned}

D x \in \mathfrak{C}_{0}, \quad x \in \mathfrak{C} .

Dx \in \text{\es {C}}_0, \quad x \in \text{\es {C}}.

x y+y x=-(x \bar{y}+y \bar{x})=-2(x, y), \quad x, y \in \mathfrak{C}_{0} .

xy + yx = -(x\overline{y} + y\overline{x}) = -2(x,y), \quad x,y \in \text{\es {C}}_0.

(D x, y)+(x, D y)=0, \quad x, y \in \mathfrak{C} .

(Dx, y) + (x, Dy) = 0, \quad x,y \in \text{\es {C}}.

\begin{array}{cll} \lambda G_{23}+\mu G_{45}+\nu G_{67}, & -\lambda G_{13}-\mu G_{46}+\nu G_{57}, & \\ \lambda G_{12}+\mu G_{47}+\nu G_{56}, & -\lambda G_{15}+\mu G_{26}-\nu G_{37}, & \lambda, \mu, \nu \in \boldsymbol{R} \\ \lambda G_{14}-\mu G_{27}-\nu G_{36}, & -\lambda G_{17}-\mu G_{24}+\nu G_{35}, & \lambda+\mu+\nu=0 \\ & \lambda G_{16}+\mu G_{25}+\nu G_{34} \end{array}

\operatorname{dim} \mathfrak{g}_{2}=14 .

\dim\text{\es {g}}_2 = 14.

\sum_{0<i<j} \lambda_{i j} F_{i j}=\sum_{0<i<j} \lambda_{i j} G_{i j} .

\sum_{0<i<j}\lambda_{ij} F_{ij} = \sum_{0<i<j}\lambda_{ij}G_{ij}.

\sum_{0<i<j} \lambda_{i j} e_{j} e_{i}=0 .

\sum_{0<i<j}\lambda_{ij}e_je_i = 0.

\begin{gathered} \lambda_{23}+\lambda_{45}+\lambda_{67}=0, \quad-\lambda_{13}-\lambda_{46}+\lambda_{57}=0, \\ \lambda_{12}+\lambda_{47}+\lambda_{56}=0, \quad-\lambda_{15}+\lambda_{26}-\lambda_{37}=0, \\ \lambda_{14}-\lambda_{27}-\lambda_{36}=0, \quad-\lambda_{17}-\lambda_{24}+\lambda_{35}=0, \\ \lambda_{16}+\lambda_{25}+\lambda_{34}=0, \end{gathered}

\boldsymbol{C}=\left\{x_{0}+x_{1} e_{1} \mid x_{i} \in \boldsymbol{R}\right\} .

\text{$C$} = \{ x_0 + x_1e_1 \, | \, x_i \in \text{$R$} \}.

\begin{aligned} x & =x_{0}+x_{1} e_{1}+x_{2} e_{2}+x_{3} e_{3}+x_{4} e_{4}+x_{5} e_{5}+x_{6} e_{6}+x_{7} e_{7} \quad\left(x_{i} \in \boldsymbol{R}\right) \\ & =\left(x_{0}+x_{1} e_{1}\right)+\left(x_{2}+x_{3} e_{1}\right) e_{2}+\left(x_{4}+x_{5} e_{1}\right) e_{4}+\left(x_{6}+x_{7} e_{1}\right) e_{6}, \end{aligned}

\begin{aligned} x \!\!\! &=& \!\!\! x_0 + x_1e_1 + x_2e_2 + x_3e_3 + x_4e_4 + x_5e_5 + x_6e_6 + x_7e_7 \quad (x_i \in \text{$R$}) \vspace{1mm}\\ \!\!\! &=& \!\!\! (x_0 + x _1e_1) + (x_2 + x_3e_1)e_2 + (x_4 + x_5e_1)e_4 + (x_6 + x_7e_1)e_6, \end{aligned}

x=a+m_{1} e_{2}+m_{2} e_{4}+m_{3} e_{6}, \quad a, m_{i} \in \boldsymbol{C} .

x = a + m_1e_2 + m_2e_4 + m_3e_6, \quad a, m_i \in \text{$C$}.

a+\left(\begin{array}{l} m_{1} \\ m_{2} \\ m_{3} \end{array}\right) .

\begin{aligned} (a+\boldsymbol{m})(b+\boldsymbol{n}) & =(a b-\langle\boldsymbol{m}, \boldsymbol{n}\rangle)+(a \boldsymbol{n}+\bar{b} \boldsymbol{m}-\overline{\boldsymbol{m} \times \boldsymbol{n}}), \\ (a+\boldsymbol{m}, b+\boldsymbol{n}) & =(a, b)+(\boldsymbol{m}, \boldsymbol{n}), \\ \overline{a+\boldsymbol{m}} & =\bar{a}-\boldsymbol{m}, \end{aligned}

\begin{aligned} (a + \text{$m$})(b + \text{$n$}) \!\!\! &=& \!\!\! (ab - \langle \text{$m$}, \text{$n$} \rangle ) + (a\text{$n$} + \overline{b}\text{$m$} - \overline{\text{$m$} \times \text{$n$}}), \vspace{1mm}\\ (a + \text{$m$}, b + \text{$n$}) \!\!\! &=& \!\!\! (a, b) + (\text{$m$}, \text{$n$}), \vspace{1mm}\\ \overline{a + \text{$m$}} \!\!\! &=& \!\!\! \overline{a} - \text{$m$}, \end{aligned}

\begin{gathered} (\boldsymbol{m}, \boldsymbol{n})=\frac{1}{2}\left(\boldsymbol{m}^{*} \boldsymbol{n}+\boldsymbol{n}^{*} \boldsymbol{m}\right)=\sum_{i=1}^{3}\left(m_{i}, n_{i}\right), \quad\langle\boldsymbol{m}, \boldsymbol{n}\rangle=\sum_{i=1}^{3} m_{i} \bar{n}_{i} \\ \boldsymbol{m} \times \boldsymbol{n}=\left(\begin{array}{c} m_{2} n_{3}-n_{2} m_{3} \\ m_{3} n_{1}-n_{3} m_{1} \\ m_{1} n_{2}-n_{1} m_{2} \end{array}\right) \end{gathered}

(\text{$m$}, \text{$n$}) = \frac{1}{2}(\text{$m$}^{*}\text{$n$} + \text{$n$}^{*}\text{$m$}) = \sum_{i=1}^3(m_i,n_i), \quad \langle \text{$m$}, \text{$n$} \rangle = \sum_{i=1}^3m_i\overline{n}_i,

\boldsymbol{C} \oplus \boldsymbol{C}^{3}=\mathfrak{C} .

\text{$C$} \oplus \text{$C$}^3 = \text{\es {C}}.

\left(\mathfrak{g}_{2}\right)_{e_{1}}=\left\{D \in \mathfrak{g}_{2} \mid D e_{1}=0\right\} .

(\text{\es {g}}_2)_{e_1} = \{ D \in \text{\es {g}}_2 \, | \, De_1 = 0 \}.

\varphi_{*}(D)(a+\boldsymbol{m})=D \boldsymbol{m}, \quad a+\boldsymbol{m} \in \boldsymbol{C} \oplus \boldsymbol{C}^{3}=\mathfrak{C} .

\varphi_*(D)(a + \text{$m$}) = D\text{$m$}, \quad a + \text{$m$} \in \text{$C$} \oplus \text{$C$}^3 = \text{\es {C}}.

\begin{array}{llll} e_{1}\left(E_{11}-E_{22}\right), & e_{1}\left(E_{22}-E_{33}\right), & E_{12}-E_{21}, & e_{1}\left(E_{12}+E_{21}\right), \\ E_{13}-E_{31}, & e_{1}\left(E_{13}+E_{31}\right), & E_{23}-E_{32}, & e_{1}\left(E_{23}+E_{32}\right) \end{array}

\begin{array}{llll} e_1(E_{11} - E_{22}), & e_1(E_{22} - E_{33}), & E_{12} - E_{21}, & e_1(E_{12} + E_{21}), \vspace{1mm}\\ E_{13} - E_{31}, & e_1(E_{13} + E_{31}), & E_{23} - E_{32}, & e_1(E_{23} + E_{32}) \end{array}

\begin{array}{ll} \varphi_{*}\left(e_{1}\left(E_{11}-E_{22}\right)\right)=-G_{23}+G_{45}, & \varphi_{*}\left(e_{1}\left(E_{22}-E_{33}\right)\right)=-G_{45}+G_{67}, \\ \varphi_{*}\left(E_{12}-E_{21}\right)=G_{24}+G_{35}, & \varphi_{*}\left(e_{1}\left(E_{12}+E_{21}\right)\right)=-G_{25}+G_{34}, \\ \varphi_{*}\left(E_{13}-E_{31}\right)=G_{26}+G_{37}, & \varphi_{*}\left(e_{1}\left(E_{13}+E_{31}\right)\right)=-G_{27}+G_{36}, \\ \varphi_{*}\left(E_{23}-E_{32}\right)=G_{46}+G_{57}, & \varphi_{*}\left(e_{1}\left(E_{23}+E_{32}\right)\right)=-G_{47}+G_{56} . \end{array}

\begin{aligned} S_{1} & =2 G_{12}-G_{47}-G_{56}, \quad S_{2}=2 G_{13}-G_{46}+G_{57}, \\ S_{3} & =2 G_{14}+G_{27}+G_{36}, \quad S_{4}=2 G_{15}+G_{26}-G_{37}, \\ S_{5} & =2 G_{16}-G_{25}-G_{34}, \quad S_{6}=2 G_{17}-G_{24}+G_{35}, \end{aligned}

\begin{array}{ll} S_1 = 2G_{12} - G_{47} - G_{56}, & S_2 = 2G_{13} - G_{46} + G_{57}, \vspace{1mm}\\ S_3 = 2G_{14} + G_{27} + G_{36}, & S_4 = 2G_{15} + G_{26} - G_{37}, \vspace{1mm}\\ S_5 = 2G_{16} - G_{25} - G_{34}, & S_6 = 2G_{17} - G_{24} + G_{35}, \end{array}

\mathfrak{g}_{2}=\mathfrak{s} \mathfrak{u}(3) \oplus \mathfrak{S} .

B=D+\sum_{i=1}^{6} x_{i} S_{i}, \quad D \in \mathfrak{s u}(3), x_{i} \in \boldsymbol{R}

B = D + \sum_{i=1}^6x_iS_i, \quad D \in \text{\es {su}}(3), x_i \in \text{$R$}.

-2 x_{1} e_{2}-2 x_{2} e_{3}-2 x_{3} e_{4}-2 x_{4} e_{5}-2 x_{5} e_{6}-2 x_{6} e_{7}=0 .

-2x_1e_2 - 2x_2e_3 - 2x_3e_4 - 2x_4e_5 - 2x_5e_6 - 2x_6e_7 = 0.

\overline{x_{1}+i x_{2}}=\bar{x}_{1}+i \bar{x}_{2}, \quad \tau\left(x_{1}+i x_{2}\right)=x_{1}-i x_{2}, \quad x_{i} \in \mathfrak{C} .

\overline{x_1 + ix_2} = \overline{x}_1 + i\overline{x}_2,\;\; \tau(x_1 + ix_2) = x_1 - ix_2, \quad x_i \in \text{\es {C}}.

\tau(x y)=(\tau x)(\tau y), \quad x, y \in \mathfrak{C}^{C} .

\tau(xy) = (\tau x)(\tau y), \quad x, y \in \text{\es {C}}^C.

\begin{gathered} H_{1}=-G_{23}+G_{45}, \quad H_{2}=-G_{45}+G_{67} \\ L_{12}=G_{24}+G_{35}, \quad L_{21}=-G_{25}+G_{34}, \quad L_{13}=G_{26}+G_{37}, \\ L_{21}=-G_{27}+G_{36}, \quad L_{23}=G_{46}+G_{57}, \quad L_{32}=-G_{47}+G_{56} . \end{gathered}

\begin{array}{c} H_1 = - G_{23}+ G_{45}, \quad H_2 = - G_{45} + G_{67} \vspace{1mm}\\ L_{12} = \;\; G_{24} + G_{35}, \quad L_{21} = - G_{25} + G_{34}, \quad L_{13} = \;\; G_{26} + G_{37}, \vspace{1mm}\\ L_{21} = - G_{27} + G_{36}, \quad L_{23} = \;\; G_{46} + G_{57}, \vspace{3mm} \quad L_{32} = - G_{47} + G_{56}. \end{array}

[\mathfrak{s u}(3), \mathfrak{S}]=\mathfrak{S} .

-x_{1} S_{6}+x_{2} S_{5} \in W \quad \cdots(\mathrm{i}) \quad \text { and } \quad-x_{1} S_{5}-x_{2} S_{6} \in W

- x_1S_6 + x_2S_5 \in W \quad \cdots \text{(i)} \quad \text{and} \quad - x_1S_5 - x_2S_6 \in W \quad \cdots \text{(ii)}

\mathfrak{g}_{2}=\mathfrak{s u}(3) \oplus \mathfrak{S} \quad \text { (Theorem 1.5.1). }

\text{\es {g}}_2 = \text{\es {su}}(3) \oplus \text{\es {S}} \quad \text{(Theorem 1.5.1)}.

\mathfrak{a} \ni\left[D^{\prime}, D+S\right]=\left[D^{\prime}, D\right]+\left[D^{\prime}, S\right], \quad\left[D^{\prime}, S\right] \in \mathfrak{S}

\text{\es {a}} \ni [D', D + S] = [D',D] + [D', S], \quad [D', S] \in \text{\es {S}}

\mathfrak{a} \supset[\mathfrak{a}, \mathfrak{S}] \supset[\mathfrak{s} \mathfrak{u}(3), \mathfrak{S}]=\mathfrak{S} \text { (Lemma 1.6.1). }

\text{\es {a}} \supset [\text{\es {a}}, \text{\es {S}}] \supset [\text{\es {su}}(3), \text{\es {S}}] = \text{\es {S}} \;\; \text{(Lemma 1.6.1)}.

\begin{array}{lll} {\left[H_{1}, L_{12}\right]=2 L_{21},} & {\left[H_{1}, L_{21}\right]=-2 L_{12},} & {\left[H_{1}, L_{13}\right]=L_{31},} \\ {\left[H_{1}, L_{31}\right]=-L_{13},} & {\left[H_{1}, L_{23}\right]=-L_{32},} & {\left[H_{1}, L_{32}\right]=L_{23} .} \end{array}

\begin{array}{llll} [H_1, L_{12}] = 2L_{21}, & [H_1, L_{21}] = - 2L_{12}, & [H_1, L_{13}] = L_{31}, \vspace{1mm}\\ {[}H_1, L_{31}] = - L_{13}, & [H_1, L_{23}] = - L_{32}, & [H_1, L_{32}] = L_{23}. \end{array}

B_{2}\left(D_{1}, D_{2}\right)=4 \operatorname{tr}\left(D_{1} D_{2}\right), \quad D_{i} \in \mathfrak{g}_{2}^{C} .

B_2(D_1, D_2) = 4\text\mathrm{{tr}}(D_1D_2), \quad D_i \in {\text{\es {g}}_2}^C.

B_{2}\left(D_{1}, D_{2}\right)=k \operatorname{tr}\left(D_{1} D_{2}\right) .

B_2(D_1, D_2) = k\text\mathrm{{tr}}(D_1D_2).

\begin{array}{ll} {\left[H_{1},\left[H_{1}, L_{12}\right]\right]=\left[H_{1}, 2 L_{21}\right]=-4 L_{12},} & {\left[H_{1},\left[H_{1}, L_{21}\right]\right]=\left[H_{1},-2 L_{12}\right]=-4 L_{21},} \\ {\left[H_{1},\left[H_{1}, L_{13}\right]\right]=\left[H_{1}, L_{31}\right]=-L_{13},} & {\left[H_{1},\left[H_{1}, L_{31}\right]\right]=\left[H_{1},-L_{13}\right]=-L_{31},} \\ {\left[H_{1},\left[H_{1}, L_{23}\right]\right]=\left[H_{1},-L_{32}\right]=-L_{23},} & {\left[H_{1},\left[H_{1}, L_{32}\right]\right]=\left[H_{1}, L_{23}\right]=-L_{32},} \end{array}

\begin{array}{ll} {\left[H_{1},\left[H_{1}, S_{1}\right]\right]=\left[H_{1}, S_{2}\right]=-S_{1},} & {\left[H_{1},\left[H_{1}, S_{2}\right]\right]=\left[H_{1},-S_{1}\right]=-S_{2},} \\ {\left[H_{1},\left[H_{1}, S_{3}\right]\right]=\left[H_{1},-S_{4}\right]=-S_{3},} & {\left[H_{1},\left[H_{1}, S_{4}\right]\right]=\left[H_{1}, S_{3}\right]=-S_{4},} \\ {\left[H_{1},\left[H_{1}, S_{5}\right]\right]=0,\left[H_{1},\left[H_{1}, S_{6}\right]\right]=0} & \end{array}

\begin{array}{ll} {[}H_1, [H_1, S_1]\,] = [H_1, S_2] = - S_1, & [H_1, [H_1, S_2]\,] = [H_1, - S_1] = - S_2, \vspace{1mm}\\ {[}H_1, [H_1, S_3]\,] = [H_1, - S_4] = - S_3, & [H_1, [H_1, S_4]\,] = [H_1, S_3] = - S_4, \vspace{1mm}\\ {[}H_1, [H_1, S_5]\,] = 0, \; [H_1, [H_1, S_6]\,] = 0 \end{array}

B_{2}\left(H_{1}, H_{1}\right)=\operatorname{tr}\left(\left(\operatorname{ad} H_{1}\right)^{2}\right)=(-4) \times 2+(-1) \times 8=-16 .

B_2(H_1, H_1) = \text\mathrm{{tr}}((\text{ad}H_1)^2) = (-4) \times 2 + (-1) \times 8 = - 16.

\begin{aligned} H_{1} H_{1} e_{2} & =H_{1} e_{3}=-e_{2}, \quad \end{aligned}

\begin{array}{ll} f_{*}(A)=\varepsilon A-\bar{\varepsilon}^{t} A, & \varepsilon=\frac{1}{2}\left(1+i e_{1}\right) \\ \varphi_{*}(D)(a+\boldsymbol{m})=D \boldsymbol{m}, & a+\boldsymbol{m} \in \boldsymbol{C}^{C} \oplus\left(\boldsymbol{C}^{3}\right)^{C}=\mathfrak{C}^{C} \end{array}

\begin{array}{l} f_*(A) = \varepsilon A - \overline{\varepsilon}\,{}^t\!A, \qquad \;\; \varepsilon = \frac{1}{2}(1 + ie_1), \vspace{1mm}\\ \varphi_*(D)(a + \text{$m$}) = D\text{$m$}, \quad a + \text{$m$} \in \text{$C$}^C \oplus (\text{$C$}^3)^C = \text{\es {C}}^C, \end{array}

\mathfrak{h}=\left\{\left.\left(\begin{array}{ccc} \lambda_{1} & 0 & 0 \\ 0 & \lambda_{2} & 0 \\ 0 & 0 & \lambda_{3} \end{array}\right) \right\rvert\, \lambda_{i} \in \boldsymbol{C}, \lambda_{1}+\lambda_{2}+\lambda_{3}=0\right\}

\text{\es {h}} = \Big\{\pmatrix{\lambda_1 & 0 & 0 \cr 0 & \lambda_2 & 0 \cr 0 & 0 & \lambda_3} \Big| \, \lambda_i \in \text{$C$}, \lambda_1 + \lambda_2 + \lambda_3 = 0 \Big\}

\pm\left(\lambda_{1}-\lambda_{2}\right), \quad \pm\left(\lambda_{1}-\lambda_{3}\right), \quad \pm\left(\lambda_{2}-\lambda_{3}\right), \quad \pm \lambda_{1}, \quad \pm \lambda_{2} \quad \pm \lambda_{3}

\pm(\lambda_1 - \lambda_2), \quad \pm(\lambda_1 - \lambda_3), \quad \pm(\lambda_2 - \lambda_3), \quad \pm\lambda_1, \quad \pm\lambda_2 \quad \pm\lambda_3

\begin{array}{ll} \pm\left(\lambda_{1}-\lambda_{2}\right) & : \pm\left(G_{24}+G_{35}\right)+i\left(-G_{25}+G_{34}\right), \\ \pm\left(\lambda_{1}-\lambda_{3}\right) & : \pm\left(G_{26}+G_{37}\right)+i\left(-G_{27}+G_{36}\right), \\ \pm\left(\lambda_{2}-\lambda_{3}\right) & : \pm\left(G_{46}+G_{57}\right)+i\left(-G_{47}+G_{56}\right) . \end{array}

\begin{aligned} \pm(\lambda_1 - \lambda_2) &:& \pm(G_{24} + G_{35}) + i(- G_{25} + G_{34}), \vspace{1mm}\\ \pm(\lambda_1 - \lambda_3) &:& \pm(G_{26} + G_{37}) + i(- G_{27} + G_{36}), \vspace{1mm}\\ \pm(\lambda_2 - \lambda_3) &:& \pm(G_{46} + G_{57}) + i(- G_{47} + G_{56}). \end{aligned}

\begin{aligned} & \pm \lambda_{1}:\left(2 G_{12}-G_{47}-G_{56}\right) \pm i\left(2 G_{13}-G_{46}+G_{57}\right), \\ & \pm \lambda_{2}:\left(2 G_{14}+G_{27}+G_{36}\right) \pm i\left(2 G_{15}+G_{26}-G_{37}\right), \\ & \pm \lambda_{3}:\left(2 G_{16}-G_{25}-G_{34}\right) \pm i\left(2 G_{17}-G_{24}+G_{35}\right) . \end{aligned}

\begin{aligned} \pm\lambda_1 &:& (2G_{12} - G_{47} - G_{56}) \pm i(2G_{13} - G_{46} + G_{57}), \vspace{1mm}\\ \pm\lambda_2 &:& (2G_{14} + G_{27} + G_{36}) \pm i(2G_{15} + G_{26} - G_{37}), \vspace{1mm}\\ \pm\lambda_3 &:& (2G_{16} - G_{25} - G_{34}) \pm i(2G_{17} - G_{24} + G_{35}). \end{aligned}

\alpha_{1}=\lambda_{1}-\lambda_{2}, \quad \alpha_{2}=\lambda_{2}

\alpha_1 = \lambda_1 - \lambda_2, \quad \alpha_2 = \lambda_2

\mu=2 \alpha_{1}+3 \alpha_{3}

\mu = 2\alpha_1 + 3\alpha_3

\begin{aligned} \lambda_{1}-\lambda_{2} & =\alpha_{1}, & \lambda_{1}-\lambda_{3} & =2 \alpha_{1}+3 \alpha_{2}, & \lambda_{2}-\lambda_{3} & =\alpha_{1}+3 \alpha_{2}, \\ \lambda_{1} & =\alpha_{1}+\alpha_{2}, & \lambda_{2} & =\alpha_{2}, & & =\lambda_{3}+2 \alpha_{2} \end{aligned}

\begin{array}{cccc} \lambda_1 - \lambda_2 = \alpha_1, & \;\;\; \lambda_1 - \lambda_3 = 2\alpha_1 + 3\alpha_2, & \lambda_2 - \lambda_3 = \alpha_1 + 3\alpha_2, \vspace{1mm}\\ \qquad \qquad \;\;\lambda_1 = \alpha_1 + \alpha_2, & \lambda_2 = \alpha_2, &\;\;\;\;\; - \lambda_3 = \alpha_1 + 2\alpha_2. \end{array}

\mathfrak{h}_{\boldsymbol{R}}=\left\{-i \lambda_{1} G_{23}-i \lambda_{2} G_{45}-i \lambda_{3} G_{67} \mid \lambda_{i} \in \boldsymbol{R}, \lambda_{1}+\lambda_{2}+\lambda_{3}=0\right\}

\text{\es {h}}_{\text{${R}$}} = \{ -i\lambda_1G_{23} -i\lambda_2G_{45} -i\lambda_3G_{67} \, |\, \lambda_i \in \text{$R$}, \lambda_1 + \lambda_2 + \lambda_3 = 0 \}

B_{2}\left(H, H^{\prime}\right)=8 \sum_{k=1}^{3} \lambda_{k} \lambda_{k}^{\prime},

B_2(H, H') = 8\sum_{k=1}^3\lambda_k{\lambda_k}',

H_{\alpha_{1}}=-\frac{1}{8} i G_{23}+\frac{1}{8} i G_{45}, \quad H_{\alpha_{2}}=\frac{1}{24} i G_{23}-\frac{1}{12} i G_{45}+\frac{1}{24} i G_{67} .

H_{\alpha_1} = - \frac{1}{8}iG_{23} + \frac{1}{8}iG_{45}, \quad H_{\alpha_2} = \frac{1}{24}iG_{23} - \frac{1}{12}iG_{45} + \frac{1}{24}iG_{67}.

\begin{aligned} & \left(\alpha_{1}, \alpha_{1}\right)=B_{2}\left(H_{\alpha_{1}}, H_{\alpha_{1}}\right)=8 \frac{1}{8} \frac{1}{8}(1+1)=\frac{1}{4} \\ & \left(\alpha_{2}, \alpha_{2}\right)=B_{2}\left(H_{\alpha_{2}}, H_{\alpha_{2}}\right)=8 \frac{1}{24} \frac{1}{24}(1+4+1)=\frac{1}{12} \\ & \left(\alpha_{1}, \alpha_{2}\right)=B_{2}\left(H_{\alpha_{1}}, H_{\alpha_{2}}\right)=8 \frac{1}{8} \frac{1}{24}(-1-2)=-\frac{1}{8} \\ & (-\mu,-\mu)=\frac{1}{4}, \quad\left(-\mu, \alpha_{1}\right)=-\frac{1}{8}, \quad\left(-\mu, \alpha_{2}\right)=0 \end{aligned}

\left(G_{2}\right)_{e_{1}}=\left\{\alpha \in G_{2} \mid \alpha e_{1}=e_{1}\right\} .

(G_2)_{e_1} = \{\alpha \in G_2 \, | \, \alpha e_1 = e_1 \}.

\varphi(A)(a+\boldsymbol{m})=a+A \boldsymbol{m}, \quad a+\boldsymbol{m} \in \boldsymbol{C} \oplus \boldsymbol{C}^{3}=\mathfrak{C} .

\varphi(A)(a + \text{$m$}) = a + A\text{$m$}, \quad a + \text{$m$} \in \text{$C$} \oplus \text{$C$}^3 = \text{\es {C}}.

\begin{aligned} (\alpha x)(\alpha y) & =(a+A \boldsymbol{m})(b+A \boldsymbol{n}) \\ & =(a b-\langle A \boldsymbol{m}, A \boldsymbol{n}\rangle)+(a A \boldsymbol{n}+\bar{b} A \boldsymbol{m}-\overline{A \boldsymbol{m} \times A \boldsymbol{n}}) \\ & =\left(a b-\left\langle\boldsymbol{m}, A^{*} A \boldsymbol{n}\right\rangle\right)+(a A \boldsymbol{n}+\bar{b} A \boldsymbol{m}-\bar{t} \widetilde{A}(\boldsymbol{m} \times \boldsymbol{n})) \\ & =(a b-\langle\boldsymbol{m}, \boldsymbol{n}\rangle)+A(a \boldsymbol{n}+\bar{b} \boldsymbol{m}-\overline{\boldsymbol{m} \times \boldsymbol{n}}) \\ & =\varphi(A)((a+\boldsymbol{m})(b+\boldsymbol{n}))=\alpha(x y) \end{aligned}

\alpha e_{2}=\boldsymbol{a}_{1}, \quad \alpha e_{4}=\boldsymbol{a}_{2}, \quad \alpha e_{6}=\boldsymbol{a}_{3}

\alpha e_2 = \text{$a$}_1, \quad \alpha e_4 = \text{$a$}_2, \quad \alpha e_6 = \text{$a$}_3

\left\langle\boldsymbol{a}_{1}, \boldsymbol{a}_{2}\right\rangle=0, \quad \boldsymbol{a}_{3}=\overline{\boldsymbol{a}_{1} \times \boldsymbol{a}_{2}} .

\langle \text{$a$}_1, \text{$a$}_2 \rangle = 0, \quad \text{$a$}_3 = \overline{\text{$a$}_1 \times \text{$a$}_2}.

a_{3}=a_{1} a_{2} .

a_3 = a_1a_2.

a_{5}=a_{1} a_{4}, \quad a_{6}=a_{4} a_{2}, \quad a_{7}=a_{3} a_{4} .

a_5 = a_1a_4, \quad a_6 = a_4a_2, \quad a_7 = a_3a_4.

\begin{aligned} & \left(a_{4}, a_{7}\right)=\left(a_{4}, a_{3} a_{4}\right)=\left(1, a_{3}\right)\left(a_{4}, a_{4}\right)=0, \\ & \left(a_{1}, a_{6}\right)=\left(a_{1}, a_{4} a_{2}\right)=-\left(a_{1} a_{2}, a_{4}\right)=-\left(a_{3}, a_{4}\right)=0, \\ & \left(a_{3}, a_{6}\right)=\left(a_{3}, a_{4} a_{2}\right)=-\left(a_{3} a_{2}, a_{4}\right)=\left(a_{1}, a_{4}\right)=0, \text { etc. } \end{aligned}

\begin{array}{l} (a_4, a_7) = (a_4, a_3a_4)= (1, a_3)(a_4, a_4) = 0, \vspace{1mm}\\ (a_1, a_6) = (a_1, a_4a_2) = -(a_1a_2, a_4) = -(a_3, a_4) = 0, \vspace{1mm}\\ (a_3, a_6) = (a_3, a_4a_2)= -(a_3a_2, a_4) = (a_1, a_4) = 0,\, \; \text{etc.} \end{array}

\alpha e_{i}=a_{i}, \quad i=0,1, \cdots, 7

\alpha e_i =a_i, \quad i = 0,1,\cdots,7

\alpha(x y)=(\alpha x)(\alpha y), \quad x, y \in \mathfrak{C} .

\alpha(xy) = (\alpha x)(\alpha y), \quad x,y \in \text{\es {C}}.

\alpha\left(e_{i} e_{j}\right)=\left(\alpha e_{i}\right)\left(\alpha e_{j}\right), \quad i, j=0,1, \cdots, 7

\alpha(e_ie_j) = (\alpha e_i)(\alpha e_j), \quad i,j = 0,1,\cdots,7

\begin{aligned} \left(\alpha e_{4}\right)\left(\alpha e_{7}\right) & =a_{4} a_{7}=a_{4}\left(a_{3} a_{4}\right)=-a_{4}\left(a_{4} a_{3}\right)=a_{3}=\alpha e_{3}=\alpha\left(e_{4} e_{7}\right), \\ \left(\alpha e_{1}\right)\left(\alpha e_{6}\right) & =a_{1} a_{6}=a_{1}\left(a_{4} a_{2}\right)=-a_{4}\left(a_{1} a_{2}\right)=-a_{4} a_{3}=a_{3} a_{4}=a_{7} \\ & =\alpha e_{7}=\alpha\left(e_{1} e_{6}\right), \\ \left(\alpha e_{3}\right)\left(\alpha e_{6}\right) & =a_{3} a_{6}=\left(a_{1} a_{2}\right)\left(a_{4} a_{2}\right)=-\left(a_{2} a_{1}\right)\left(a_{4} a_{2}\right)=-a_{2}\left(a_{1} a_{4}\right) a_{2} \\ & =-a_{2} a_{5} a_{2}=a_{2} a_{2} a_{5}=-a_{5}=-\alpha e_{5}=\alpha\left(e_{3} e_{6}\right), \text { etc. } \end{aligned}

\begin{aligned} (\alpha e_4)(\alpha e_7) \!\!\! &=& \!\!\! a_4 a_7 = a_4(a_3 a_4) = - a_4(a_4a_3) = a_3 = \alpha e_3 = \alpha (e_4e_7), \vspace{1mm}\\ (\alpha e_1)(\alpha e_6) \!\!\! &=& \!\!\! a_1 a_6 = a_1 (a_4 a_2)= - a_4(a_1a_2) = - a_4 a_3 = a_3a_4 = a_7 \vspace{1mm}\\ \!\!\! &=& \!\!\! \alpha e_7 = \alpha(e_1e_6),\\ (\alpha e_3)(\alpha e_6) \!\!\! &=& \!\!\! a_3a_6 = (a_1a_2)(a_4a_2) = -(a_2 a_1)(a_4 a_2) = - a_2(a_1a_4)a_2 \vspace{1mm}\\ \!\!\! &=& \!\!\! - a_2a_5a_2 = a_2a_2a_5 = - a_5 = - \alpha e_5 = \alpha(e_3e_6), \,\; \text{etc.} \end{aligned}

w=\varphi\left(\operatorname{diag}\left(\omega_{1}, \omega_{1}, \omega_{1}\right)\right)

w = \varphi(\text\mathrm{{diag}}(\omega_1, \omega_1, \omega_1))

w(a+\boldsymbol{m})=a+\omega_{1} \boldsymbol{m}, \quad a+\boldsymbol{m} \in \boldsymbol{C} \oplus \boldsymbol{C}^{3}=\mathfrak{C} .

w(a + \text{$m$}) = a + \omega_1\text{$m$}, \quad a + \text{$m$} \in \text{$C$} \oplus \text{$C$}^3 = \text{\es {C}}.

\left(G_{2}\right)^{w}=\left\{\alpha \in G_{2} \mid w \alpha=\alpha w\right\} .

(G_2)^w = \{ \alpha \in G_2 \, | \, w\alpha = \alpha w \}.

\begin{aligned} w \varphi(A)(a+\boldsymbol{m}) & =w(a+A \boldsymbol{m})=a+\omega_{1} A \boldsymbol{m} \\ & =a+A \omega_{1} \boldsymbol{m}=\varphi(A) w(a+\boldsymbol{m}) . \end{aligned}

\begin{aligned} w\varphi(A)(a + \text{$m$}) \!\!\! &=& \!\!\! w(a + A\text{$m$}) = a + \omega_1A\text{$m$} \\ \!\!\! &=& \!\!\! a + A\omega_1\text{$m$} = \varphi(A)w(a + \text{$m$}). \end{aligned}

\alpha e_{1}=e_{1} \quad \text { or } \quad \alpha e_{1}=-e_{1} .

\alpha e_1 = e_1 \quad \text{or} \quad \alpha e_1 = - e_1.

\omega_{1} \overline{\boldsymbol{m}}=w\left(\gamma_{1} \boldsymbol{m}\right)=\gamma_{1}(w \boldsymbol{m})=\overline{\omega_{1} \boldsymbol{m}}=\overline{\omega_{1}} \overline{\boldsymbol{m}} \quad \text { for all } \boldsymbol{m} \in \boldsymbol{C}^{3},

\omega_1\overline{\text{$m$}} = w(\gamma_1\text{$m$}) = \gamma_1(w\text{$m$}) = \overline{\omega_1\text{$m$}} = \overline{\omega}_1\overline{\text{$m$}} \quad \text{for all $\text{$m$} \in \text{$C$}^3$,}

\boldsymbol{H}=\left\{x_{0}+x_{1} e_{1}+x_{2} e_{2}+x_{3} e_{3} \mid x_{i} \in \boldsymbol{R}\right\} .

\text{$H$} = \{ x_0 + x_1e_1 + x_2e_2 + x_3e_3 \, | \, x_i \in \text{$R$} \}.

\begin{aligned} x & =x_{0}+x_{1} e_{1}+x_{2} e_{2}+x_{3} e_{3}+x_{4} e_{4}+x_{5} e_{5}+x_{6} e_{6}+x_{7} e_{7} \quad\left(x_{i} \in \boldsymbol{R}\right) \\ & =\left(x_{0}+x_{1} e_{1}+x_{2} e_{2}+x_{3} e_{3}\right)+\left(x_{4}+x_{5} e_{1}-x_{6} e_{2}+x_{7} e_{3}\right) e_{4}, \end{aligned}

\begin{aligned} x \!\!\! &=& \!\!\! x_0 + x_1e_1 + x_2e_2 + x_3e_3 + x_4e_4 + x_5e_5 + x_6e_6 + x_7e_7 \quad (x_i \in \text{$R$}) \vspace{1mm}\\ \!\!\! &=& \!\!\! (x_0 + x _1e_1 + x_2e_2 + x_3e_3) + (x_4 + x_5e_1 - x_6e_2 + x_7e_3)e_4, \end{aligned}

x=m+a e_{4}, \quad m, a \in \boldsymbol{H} .

x = m + ae_4, \quad m,a \in \text{$H$}.

\begin{aligned} \left(m+a e_{4}\right)\left(n+b e_{4}\right) & =(m n-\bar{b} a)+(a \bar{n}+b m) e_{4}, \\ \left(m+a e_{4}, n+b e_{4}\right) & =(m, n)+(a, b), \\ \overline{m+a e_{4}} & =\bar{m}-a e_{4}, \\ \gamma\left(m+a e_{4}\right) & =m-a e_{4} . \end{aligned}

\begin{aligned} (m + ae_4)(n + be_4) \!\!\! &=& \!\!\! (mn - \overline{b}a) + (a\overline{n} + bm)e_4,\\ (m + ae_4, n + be_4) \!\!\! &=& \!\!\! (m, n)+(a, b),\\ \overline{m + ae_4} \!\!\! &=& \!\!\! \overline{m} - ae_4, \\ \gamma(m + ae_4) \!\!\! &=& \!\!\! m - ae_4. \end{aligned}

\boldsymbol{H} \oplus \boldsymbol{H} e_{4}=\mathfrak{C} .

\text{$H$} \oplus \text{$H$} e_4 = \text{\es {C}}.

\left(G_{2}\right)^{\gamma}=\left\{\alpha \in G_{2} \mid \gamma \alpha=\alpha \gamma\right\} .

(G_2)^{\gamma} = \{ \alpha \in G_2 \, | \, \gamma \alpha =\alpha \gamma \}.

\varphi(p, q)\left(m+a e_{4}\right)=q m \bar{q}+(p a \bar{q}) e_{4}, \quad m+a e_{4} \in \boldsymbol{H} \oplus \boldsymbol{H} e_{4}=\mathfrak{C} .

\varphi(p,q)(m + ae_4) = qm\overline{q} + (pa\overline{q})e_4, \quad m + ae_4 \in \text{$H$} \oplus \text{$H$} e_4 = \text{\es {C}}.

\begin{aligned} (\alpha x)(\alpha y) & =\left(q m \bar{q}+(p a \bar{q}) e_{4}\right)\left(q n \bar{q}+(p b \bar{q}) e_{4}\right) \\ & =((q m \bar{q})(q n \bar{q})-(\overline{p b \bar{q}})(p a \bar{q}))+((p a \bar{q})(\overline{q n \bar{q}})+(p b \bar{q})(q m \bar{q})) e_{4} \\ & =q(m n-\bar{b} a) \bar{q}+(p(a \bar{n}+b m) \bar{q}) e_{4} \\ & =\varphi(p, q)\left(\left(m+a e_{4}\right)\left(n+b e_{4}\right)\right)=\alpha(x y) \end{aligned}

\alpha m=q m \bar{q}, \quad m \in \boldsymbol{H}

\alpha m = qm\overline{q}, \quad m \in \text{$H$}

\beta\left(m+a e_{4}\right)=\beta m+(\beta a)\left(\beta e_{4}\right)=m+a\left(p e_{4}\right)=m+(p a) e_{4}=\varphi(p, 1)\left(m+a e_{4}\right),

\beta(m + ae_4) = \beta m + (\beta a)(\beta e_4) = m + a(pe_4) = m + (pa)e_4 = \varphi(p,1)(m + ae_4),

\alpha=\varphi(1, q) \beta=\varphi(1, q) \varphi(p, 1)=\varphi(p, q), \quad(p, q) \in \operatorname{Sp}(1) \times \operatorname{Sp}(1),

\alpha =\varphi(1,q)\beta = \varphi(1,q)\varphi(p,1) = \varphi(p,q), \quad (p, q) \in Sp(1) \times Sp(1),

f(p, q) x=p x \bar{q}, \quad x \in \boldsymbol{H}

f(p, q)x = px\overline{q}, \quad x \in \text{$H$}

\begin{array}{cc} 2 G_{12}-G_{47}-G_{56}, & -G_{47}+G_{56}, \\ 2 G_{13}-G_{46}-G_{57}, & G_{46}+G_{57}, \\ 2 G_{23}-G_{45}-G_{67}, & -G_{45}+G_{67} \end{array}

\begin{array}{ll} 2G_{12} - G_{47} - G_{56}, & - G_{47} + G_{56}, \vspace{1mm}\\ 2G_{13} - G_{46} - G_{57}, & \;\;\; G_{46} + G_{57}, \vspace{1mm}\\ 2G_{23} - G_{45} - G_{67}, & - G_{45} + G_{67} \end{array}

z\left(G_{2}\right)=\{1\} .

z(G_2)=\{1\}.

\alpha=1 \quad \text { or } \quad \alpha=\gamma .

\alpha = 1 \quad \text{or} \quad \alpha =\gamma.

G_{2}{ }^{C}=\left\{\alpha \in \operatorname{Iso}_{C}\left(\mathfrak{C}^{C}\right) \mid \alpha(x y)=(\alpha x)(\alpha y)\right\} .

{G_2}^C = \{ \alpha \in \text\mathrm{{Iso}}_C(\text{\es {C}}^C) \, | \, \alpha(xy) = (\alpha x)(\alpha y) \}.

(\alpha x, \alpha y)=(x, y), \quad x, y \in \mathfrak{C}^{C} .

(\alpha x, \alpha y) = (x, y), \quad x, y \in \text{\es {C}}^C.

\overline{\alpha e_{k}}=-\alpha e_{k}, \quad k=1,2, \cdots, 7 .

\overline{\alpha e_k} = - \alpha e_k, \quad k = 1, 2, \cdots, 7.

\overline{\alpha x}=\alpha \bar{x}, \quad x \in \mathfrak{C}^{C} .

\overline{\alpha x}=\alpha \overline{x}, \quad x \in \text{\es {C}}^C.

\begin{aligned} (\alpha x, \alpha y) & =\frac{1}{2}((\alpha x)(\overline{\alpha y})+(\alpha y)(\overline{\alpha x}))=\frac{1}{2}((\alpha x)(\alpha \bar{y})+(\alpha y)(\alpha \bar{x})) \\ & =\alpha\left(\frac{1}{2}(\alpha(x \bar{y}+y \bar{x}))=\alpha((x, y))=(x, y) .\right. \end{aligned}

\begin{aligned} (\alpha x, \alpha y) \!\!\! &=& \!\!\! \frac{1}{2}((\alpha x)(\overline{\alpha y}) + (\alpha y)(\overline{\alpha x}))=\frac{1}{2}((\alpha x)(\alpha \overline{y}) + (\alpha y)(\alpha \overline{x})) \vspace{1mm}\\ \!\!\! &=& \!\!\! \alpha\Big(\frac{1}{2}(\alpha(x\overline{y} + y\overline{x})\Big) = \alpha((x,y)) = (x,y). \end{aligned}

\langle x, y\rangle=(\tau x, y),

\langle x, y \rangle = (\tau x, y),

G_{2}=\left\{\alpha \in G_{2}^{C} \mid \tau \alpha=\alpha \tau\right\} .

G_2 = \{ \alpha \in {G_2}^C \, | \, \tau\alpha = \alpha\tau \}.

G_{2}{ }^{C} \simeq G_{2} \times \boldsymbol{R}^{14} .

{G_2}^C \simeq G_2 \times \text{$R$}^{14}.

G_{2}{ }^{C} \simeq\left(G_{2}{ }^{C} \cap U\left(\mathfrak{C}^{C}\right)\right) \times \boldsymbol{R}^{d}=G_{2} \times \boldsymbol{R}^{d},

{G_2}^C \simeq ({G_2}^C \cap U(\text{\es {C}}^C)) \times \text{$R$}^d = G_2 \times \text{$R$}^d,

\left(m+a e_{4}{ }^{\prime}\right)\left(n+b e_{4}{ }^{\prime}\right)=(m n+\bar{b} a)+(a \bar{n}+b m) e_{4}{ }^{\prime} .

(m + a{e_4}')(n + b{e_4}') = (mn + \overline{b}a) + (a\overline{n} + bm){e_4}'.

G_{2(2)}=\left\{\alpha \in \operatorname{Iso}_{\boldsymbol{R}}\left(\mathfrak{C}^{\prime}\right) \mid \alpha(x y)=(\alpha x)(\alpha y)\right\} .

G_{2(2)} = \{ \alpha \in \text\mathrm{{Iso}}_{\text{${R}$}}(\text{\es {C}}') \, | \, \alpha(xy) = (\alpha x)(\alpha y) \}.

G_{2(2)}=\left(G_{2}{ }^{C}\right)^{\tau \gamma}=\left\{\alpha \in G_{2}{ }^{C} \mid \tau \gamma \alpha=\alpha \gamma \tau\right\} .

G_{2(2)} = ({G_2}^C)^{\tau\gamma} = \{ \alpha \in {G_2}^C \, |\, \tau\gamma\alpha = \alpha\gamma\tau \}.

G_{2(2)} \simeq(S p(1) \times S p(1)) / \boldsymbol{Z}_{2} \times \boldsymbol{R}^{8} .

G_{2(2)} \simeq (Sp(1) \times Sp(1))/\text{$Z$}_2 \times \text{$R$}^{8}.

\begin{aligned} G_{2(2)} & \simeq\left(G_{2(2)} \cap O(8)\right) \times \boldsymbol{R}^{d} \\ & =\left(\left(G_{2}{ }^{C}\right)^{\tau \gamma}\right)^{\gamma} \times \boldsymbol{R}^{d}=\left(\left(G_{2}{ }^{C}\right)^{\tau}\right)^{\gamma} \times \boldsymbol{R}^{d}=\left(G_{2}\right)^{\gamma} \times \boldsymbol{R}^{d} \\ & =(\operatorname{Sp}(1) \times \operatorname{Sp}(1)) / \boldsymbol{Z}_{2} \times \boldsymbol{R}^{d}(\text { Theorem 1.10.1 }), \quad d=8 . \end{aligned}

\begin{aligned} G_{2(2)} \!\!\! &\simeq& \!\!\! (G_{2(2)} \cap O(8)) \times \text{$R$}^d \vspace{1mm}\\ \!\!\! &=& \!\!\! (({G_2}^C)^{\tau\gamma})^{\gamma} \times \text{$R$}^d = (({G_2}^C)^{\tau})^{\gamma} \times \text{$R$}^d = (G_2)^{\gamma} \times \text{$R$}^d \vspace{1mm}\\ \!\!\! &=& \!\!\! (Sp(1) \times Sp(1))/\text{$Z$}_2 \times \text{$R$}^d \;\, \text{(Theorem 1.10.1)},\;\; d = 8. \end{aligned}

z\left(G_{2(2)}\right)=\{1\} .

z(G_{2(2)}) = \vspace{4mm} \{1\}.

D_{a} x=x-2(x, a) a, \quad x \in \boldsymbol{R}^{n} .

D_ax = x - 2(x, a)a, \quad x \in \text{$R$}^n.

A=D_{a_{m}} \cdots D_{a_{2}} D_{a_{1}}, \quad a_{i} \in S^{n-1}

A = D_{a_m} \cdots D_{a_2}D_{a_1}, \quad a_i \in S^{n-1}.

A=D_{a_{2 m}} \cdots D_{a_{2}} D_{a_{1}}, \quad a_{i} \in S^{n-1}

A = D_{a_{2m}}\cdots D_{a_2}D_{a_1}, \quad a_i \in S^{n-1}.

S O(\mathfrak{C})=\left\{\alpha \in \operatorname{Iso}_{\boldsymbol{R}}(\mathfrak{C}) \mid(\alpha x, \alpha y)=(x, y), \operatorname{det} \alpha=1\right\} .

SO(\text{\es {C}}) = \{ \alpha \in \text\mathrm{{Iso}}_{\text{${R}$}}(\text{\es {C}}) \,| \, (\alpha x,\alpha y) = (x,y), \text\mathrm{{det}} \alpha = 1 \}.

\left(\alpha_{1} x\right)\left(\alpha_{2} y\right)=\alpha_{3}(x y), \quad x, y \in \mathfrak{C} .

(\alpha_1 x)(\alpha_2 y) = \alpha_3 (xy), \quad x,y \in \text{\es {C}}.

D_{a} x=x-2(x, a) a=x-(x \bar{a}+a \bar{x}) a=-a \bar{x} a, \quad x \in \mathfrak{C},

D_a x = x - 2(x,a)a = x - (x\overline{a} + a\overline{x})a = -a\overline{x}a, \quad x \in \text{\es {C}},

\left(\alpha_{1} x\right)\left(\alpha_{2} y\right)=(b(\bar{a} x))((y \bar{a}) b)=b(\bar{a}(x y) \bar{a}) b=\alpha_{3}(x y), \quad x, y \in \mathfrak{C} .

(\alpha_1x)(\alpha_2y) = (b(\overline{a}x))((y\overline{a})b) = b(\overline{a}(xy)\overline{a})b = \alpha_3(xy), \quad x,y \in \text{\es {C}}.

\left(\alpha_{1} x\right)\left(\alpha_{2} y\right)=x y, \quad x, y \in \mathfrak{C} .

(\alpha_1x)(\alpha_2y) = xy, \quad x,y \in \text{\es {C}}.

(x p)(\bar{p} y)=x y, \quad x, y \in \mathfrak{C} .

(xp)(\bar{p}y) = xy, \quad x, y \in \text{\es {C}}.

(x p) y=x(p y), \quad x, y \in \mathfrak{C} .

(xp)y = x(py), \quad x,y \in \text{\es {C}}.

\left(\alpha_{1} x\right)\left(\alpha_{2} y\right)=\overline{\alpha_{3}(\overline{x y})}, \quad x, y \in \mathfrak{C}

\begin{aligned} (\alpha_1x)(\alpha_2y) \!\! &=& \!\! \overline{\alpha_3(\overline{xy})}, \quad x,y \in \text{\es {C}} \end{aligned}

\begin{array}{ll} \left(\alpha_{2} x\right)\left(\alpha_{3} y\right)=\overline{\alpha_{1}(\overline{x y})}, & x, y \in \mathfrak{C}, \\ \left(\alpha_{3} x\right)\left(\alpha_{1} y\right)=\overline{\alpha_{2}(\overline{x y})}, & x, y \in \mathfrak{C} . \end{array}

\begin{aligned} (\alpha_2x)(\alpha_3y) \!\! &=& \!\! \overline{\alpha_1(\overline{xy})}, \quad x,y \in \text{\es {C}},\\ (\alpha_3x)(\alpha_1y) \!\! &=& \!\! \overline{\alpha_2(\overline{xy})}, \quad x,y \in \text{\es {C}}. \end{aligned}

\left(\alpha_{2} y\right)\left(\alpha_{3}(\overline{x y})\right)=\overline{\alpha_{1} x}|y|^{2} .

(\alpha_2y)(\alpha_3(\overline{xy})) = \overline{\alpha_1x}|y|^2.

\left(\alpha_{2} y\right)\left(\alpha_{3} z\right)=\overline{\alpha_{1}(\overline{y z})} .

(\alpha_2y)(\alpha_3z) = \overline{\alpha_1(\overline{yz})}.

\left(\alpha_{1} x\right)\left(\alpha_{2} y\right)=\alpha_{3}(x y), \quad x, y \in \mathfrak{C},

(\alpha_1x)(\alpha_2y) = \alpha_3(xy), \quad x,y \in \text{\es {C}},

\left(\beta_{1}\left(\alpha_{1} x\right)\right)\left(\beta_{2}\left(\alpha_{2} y\right)\right)=\beta_{3}\left(\left(\alpha_{1} x\right)\left(\alpha_{2} y\right)\right)=\beta_{3}\left(\alpha_{3}(x y)\right), \quad x, y \in \mathfrak{C} .

(\beta_1(\alpha_1x))(\beta_2(\alpha_2y)) = \beta_3((\alpha_1x)(\alpha_2 y)) = \beta_3(\alpha_3(xy)), \quad x,y \in \text{\es {C}} .

\bar{x}\left(\gamma_{2} y\right)=\gamma_{3}(x y), \quad x, y \in \mathfrak{C} .

\overline{x}(\gamma_2y) = \gamma_3(xy), \quad x,y \in \text{\es {C}}.

\bar{x}\left(\gamma_{2} y\right)=\gamma_{2}(x y), \quad x, y \in \mathfrak{C} .

\displaylines{\hfill \overline{x}(\gamma_2y) = \gamma_2(xy), \quad x,y \in \text{\es {C}}. \hfill\text{(i)}}

\bar{x}(\bar{y} p)=(\overline{x y}) p, \quad x, y \in \mathfrak{C} .

\displaylines{\hfill \overline{x}(\overline{y}p) = (\overline{xy})p, \quad x,y \in \text{\es {C}}. \hfill\text{(ii)}}

p x=x p \quad \text { for all } x \in \mathfrak{C} .

px = xp \quad \text{for all}\;\; x \in \text{\es {C}}.

x y=y x \quad \text { for all } x, y \in \mathfrak{C} .

xy = yx \quad \text{for all}\;\; x,y \in \text{\es {C}}.

\alpha_{a} x=a x a^{-1}, \quad x \in \mathfrak{C}

\alpha_ax = axa^{-1}, \quad x \in \text{\es {C}}

(x \bar{a})(a y a)=(x y) a, \quad x, y \in \mathfrak{C} .

(x\overline{a})(aya) = (xy)a, \quad x, y \in \text{\es {C}}.

(a x \bar{a})\left(a y a^{2}\right)=a(x y) a^{2}, \quad x, y \in \mathfrak{C} .

(ax\overline{a})(aya^2) = a(xy)a^2, \quad x, y \in \text{\es {C}}.

(a x \bar{a})(a y \bar{a})=a(x y) \bar{a}, \quad x, y \in \mathfrak{C} .

(ax\overline{a})(ay\overline{a}) = a(xy)\overline{a}, \quad x, y \in \text{\es {C}}.

a y a^{2}= \pm a y \bar{a} .

aya^2 = \pm ay\overline{a}.

\alpha_{\bar{\omega}_{1}}(a+\boldsymbol{m})=\bar{\omega}_{1}(a+\boldsymbol{m}) \omega_{1}=\bar{\omega}_{1} a \omega_{1}+\bar{\omega}_{1}^{2} \boldsymbol{m}=a+\omega_{1} \boldsymbol{m}=w(a+\boldsymbol{m}),

\alpha_{\overline{\omega}_1}(a + \text{$m$}) = \overline{\omega}_1(a + \text{$m$})\omega_1 = \overline{\omega}_1a\omega_1 + {\overline{\omega}_1}^2\text{$m$} = a + \omega_1\text{$m$} = w(a + \text{$m$}),

(\alpha x)(\widetilde{\alpha} y)=\alpha^{\prime}(x y), \quad x, y \in \mathfrak{C} .

\displaylines{\hfill (\alpha x)(\widetilde{\alpha}y) = \alpha'(xy), \quad x, y \in \text{\es {C}}. \hfill\text{(i)}}

(\alpha x)(\widetilde{\alpha} y)=\widetilde{\alpha}(x y), \quad x, y \in \mathfrak{C} .

\displaylines{\hfill (\alpha x)(\widetilde{\alpha}y) = \widetilde{\alpha}(xy),\quad x, y \in \text{\es {C}}. \hfill\text{(ii)}}

\widetilde{B}_{3}=\{\widetilde{\alpha} \in S O(8) \mid(\alpha x)(\widetilde{\alpha} y)=\widetilde{\alpha}(x y), x, y \in \mathfrak{C} \text { for some } \alpha \in S O(7)\} .

\widetilde{B}_3 = \{\widetilde{\alpha} \in SO(8) \, | \, (\alpha x)(\widetilde{\alpha}y) = \widetilde{\alpha}(xy), x,y \in \text{\es {C}} \, \, \, \text{for some }\; \alpha \in SO(7)\}.

a_{5} b_{0}=a_{1}\left(a_{4} b_{0}\right), \quad a_{6} b_{0}=a_{2}\left(a_{4} b_{0}\right), \quad a_{7} b_{0}=a_{6}\left(a_{1} b_{0}\right) .

a_5b_0 = a_1(a_4b_0), \quad a_6b_0 = a_2(a_4b_0), \quad a_7b_0 = a_6(a_1b_0).

\begin{aligned} \left(a_{2}, a_{6}\right) & =\left(a_{2} b_{0}, a_{6} b_{0}\right)=\left(a_{2} b_{0}, a_{2}\left(a_{4} b_{0}\right)\right)=\left(b_{0}, a_{4} b_{0}\right)=\left(1, a_{4}\right)=0, \\ \left(a_{3}, a_{7}\right) & =\left(a_{3} b_{0}, a_{7} b_{0}\right)=\left(a_{2}\left(a_{1} b_{0}\right), a_{6}\left(a_{1} b_{0}\right)\right)=\left(a_{1}\left(a_{2} b_{0}\right), a_{1}\left(a_{6} b_{0}\right)\right) \\ & =\left(a_{2} b_{0}, a_{6} b_{0}\right)=\left(a_{2} b_{0}, a_{2}\left(a_{4} b_{0}\right)\right)=\left(b_{0}, a_{4} b_{0}\right)=\left(1, a_{4}\right)=0 . \end{aligned}

\begin{aligned} (a_2, a_6) \!\!\! &=& \!\!\! (a_2b_0, a_6b_0)=(a_2b_0, a_2(a_4b_0)) = (b_0, a_4b_0) = (1, a_4) = 0, \vspace{1mm}\\ (a_3, a_7) \!\!\! &=& \!\!\! (a_3b_0, a_7b_0) = (a_2(a_1b_0), a_6(a_1b_0)) = (a_1(a_2b_0), a_1(a_6b_0))\\ \!\!\!&=&\!\!\! (a_2b_0, a_6b_0) = (a_2b_0, a_2(a_4b_0)) = (b_0, a_4b_0) = (1, a_4) = 0. \end{aligned}

\alpha e_{i}=a_{i}, \quad i=0,1, \cdots, 7

\alpha e_i = a_i, \quad i = 0, 1, \cdots, 7

(\alpha x)\left((\alpha y) b_{0}\right)=(\alpha(x y)) b_{0}, \quad x, y \in \mathfrak{C} .

\displaylines{\hfill (\alpha x)((\alpha y)b_0)=(\alpha(xy))b_0, \quad x,y \in \text{\es {C}}. \hfill\text{(i)}}

\left(\alpha e_{i}\right)\left(\left(\alpha e_{j}\right) b_{0}\right)=\left(\alpha\left(e_{i} e_{j}\right)\right) b_{0}, \quad i, j=0,1, \cdots, 7 .

(\alpha e_i)((\alpha e_j)b_0) = (\alpha(e_ie_j))b_0, \quad i, j = 0, 1, \cdots, 7.

\begin{aligned} \left(\alpha e_{1}\right)\left(\left(\alpha e_{3}\right) b_{0}\right) & =a_{1}\left(a_{3} b_{0}\right)=a_{1}\left(a_{2}\left(a_{1} b_{0}\right)\right)=-a_{1}\left(a_{1}\left(a_{2} b_{0}\right)\right)=a_{2} b_{0} \\ & =\alpha\left(e_{2}\right) b_{0}=\alpha\left(e_{1} e_{3}\right) b_{0}, \\ \left(\alpha e_{2}\right)\left(\left(\alpha e_{5}\right) b_{0}\right) & =a_{2}\left(a_{5} b_{0}\right)=a_{2}\left(a_{1}\left(a_{4} b_{0}\right)\right)=-a_{1}\left(a_{2}\left(a_{4} b_{0}\right)\right)=-a_{1}\left(a_{6} b_{0}\right) \\ & =a_{6}\left(a_{1} b_{0}\right)=a_{7} b_{0}=\alpha\left(e_{7}\right) b_{0}=\alpha\left(e_{2} e_{5}\right) b_{0} . \end{aligned}

\begin{aligned} (\alpha e_1)((\alpha e_3)b_0) \!\!\! &=& \!\!\! a_1(a_3b_0) = a_1(a_2(a_1b_0)) = - a_1(a_1(a_2b_0)) = a_2b_0 \vspace{1mm}\\ \!\!\! &=& \!\!\! \alpha(e_2)b_0 = \alpha(e_1e_3)b_0, \vspace{1mm}\\ (\alpha e_2)((\alpha e_5)b_0) \!\!\! &=& \!\!\! a_2(a_5b_0) = a_2(a_1(a_4b_0)) = - a_1(a_2(a_4b_0)) = - a_1(a_6b_0) \vspace{1mm}\\ \!\!\! &=& \!\!\! a_6(a_1b_0) = a_7b_0 = \alpha(e_7)b_0 = \alpha(e_2e_5)b_0. \end{aligned}

b_{i}=a_{i} b_{0}, \quad i=0,1, \cdots, 7,

b_i = a_ib_0, \quad i = 0, 1, \cdots, 7,

\widetilde{\alpha} e_{i}=b_{i}, \quad i=0,1, \cdots, 7

\widetilde{\alpha}e_i=b_i, \quad i = 0, 1, \cdots, 7

(\alpha x)(\widetilde{\alpha} y)=\widetilde{\alpha}(x y), \quad x, y \in \mathfrak{C} .

\displaylines{\hfill (\alpha x)(\widetilde{\alpha}y) = \widetilde{\alpha}(xy), \quad x, y \in \text{\es {C}}. \hfill\text{(ii)}}

\widetilde{B}_{3} \cong \operatorname{Spin}(7) .

(\alpha x)(\widetilde{\alpha} y)=\widetilde{\alpha}(x y), \quad x, y \in \mathfrak{C} .

(\alpha x)(\widetilde{\alpha} y) = \widetilde{\alpha}(xy), \quad x, y \in \text{\es {C}}.

\alpha(x y)=(\widetilde{\alpha} x) \overline{(\widetilde{\alpha} \bar{y})}, \quad x, y \in \mathfrak{C} .

\alpha(xy) = (\widetilde{\alpha}x)\overline{(\widetilde{\alpha}\overline{y})}, \quad x,y \in \text{\es {C}}.

\widetilde{B}_{3} /\{1,-1\} \cong S O(7) .

\widetilde{B}_3/\{1, -1 \} \cong SO(7).

\begin{aligned} & \widetilde{D}_{4}=\left\{\left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right) \in S O(8) \times S O(8) \times S O(8) \mid\left(\alpha_{1} x\right)\left(\alpha_{2} y\right)=\overline{\alpha_{3}(\overline{x y})}, x, y \in \mathfrak{C}\right\} \\ & \widetilde{D}_{4} \text { is a compact group. } \end{aligned}

\widetilde{D}_4 = \{(\alpha_1, \alpha_2, \alpha_3) \in SO(8) \times SO(8) \times SO(8) \, | \, (\alpha_1x)(\alpha_2y) = \overline{\alpha_3(\overline{xy})}, x,y \in \text{\es {C}} \}.

\operatorname{Spin}(7) \ni \widetilde{\alpha} \longleftrightarrow(\alpha, \widetilde{\alpha}, \kappa \widetilde{\alpha}) \in \widetilde{D}_{4} .

Spin(7) \ni \widetilde{\alpha} \;\longleftrightarrow \; (\alpha,\widetilde{\alpha},\kappa \widetilde{\alpha}) \vspace{3mm} \in \widetilde{D}_4.

\widetilde{D}_{4} / \operatorname{Spin}(7) \simeq S^{7} .

\left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right) a=\alpha_{1} a, \quad a \in S^{7} .

(\alpha_1, \alpha_2, \alpha_3)a = \alpha_1a, \quad a \in S^7.

\left(\alpha_{1} x\right)\left(\alpha_{2} y\right)=\overline{\alpha_{3}(\overline{x y})} . \quad x, y \in \mathfrak{C} .

(\alpha_1x)(\alpha_2y) = \overline{\alpha_3(\overline{xy})}. \quad x, y \in \text{\es {C}}.

\widetilde{D}_{4} \cong \operatorname{Spin}(8) .

p\left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right)=\alpha_{1} .

p(\alpha_1, \alpha_2, \alpha_3) = \alpha_1.

\widetilde{D}_{4} /\{(1,1,1),(1,-1,-1)\} \cong S O(8) .

\widetilde{D}_4/\{(1,1,1),(1,-1,-1)\} \cong SO(8).

\begin{aligned} z(\operatorname{Spin}(8)) & =\{(1,1,1),(1,-1,-1),(-1,-1,1),(-1,1,-1)\} \\ & =\{(1,1,1),(1,-1,-1)\} \times\{(1,1,1),(-1,-1,1)\} \cong \boldsymbol{Z}_{2} \times \boldsymbol{Z}_{2} . \end{aligned}

\begin{aligned} z(Spin(8)) \!\!\! &=& \!\!\! \{(1,1,1),(1,-1,-1),(-1,-1,1),(-1,1,-1) \}\\ \!\!\! &=& \!\!\! \{(1,1,1),(1,-1,-1) \} \times \{(1,1,1),(-1,-1,1) \} \cong \text{$Z$}_2 \times \text{$Z$}_2. \end{aligned}

\begin{aligned} & \kappa\left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right)=\left(\kappa \alpha_{1}, \kappa \alpha_{3}, \kappa \alpha_{2}\right), \\ & \pi\left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right)=\left(\kappa \alpha_{3}, \kappa \alpha_{2}, \kappa \alpha_{1}\right), \\ & \nu\left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right)=\left(\alpha_{2}, \alpha_{3}, \alpha_{1}\right), \end{aligned}

\kappa^{2}=1, \quad \pi^{2}=1, \quad \nu^{3}=1, \quad \nu=\pi \kappa .

\kappa^2 = 1, \quad \pi^2 = 1, \quad \nu^3 = 1, \quad \nu = \pi\kappa.

\begin{aligned} \operatorname{Spin}(7) & =\{\alpha \in \operatorname{Spin}(8) \mid \kappa \alpha=\alpha\}, \\ G_{2} & =\left\{\alpha \in \operatorname{Spin}(8) \mid \lambda \alpha=\alpha, \lambda \in \mathfrak{S}_{3}\right\} \\ & =\{\alpha \in \operatorname{Spin}(8) \mid \pi \alpha=\alpha, \nu \alpha=\alpha\} \\ & =\{\alpha \in \operatorname{Spin}(8) \mid \nu \alpha=\alpha\} \\ & =\{\alpha \in \operatorname{Spin}(7) \mid \pi \alpha=\alpha\} . \end{aligned}

\begin{aligned} Spin(7) \!\!\! &=& \!\!\! \{ \alpha \in Spin(8) \, | \, \kappa\alpha = \alpha \},\\ G_2 \!\!\! &=& \!\!\! \{ \alpha \in Spin(8) \,| \, \lambda\alpha = \alpha, \lambda \in \text{\es {S}}_3 \}\\ \!\!\! &=& \!\!\! \{ \alpha \in Spin(8) \,| \, \pi\alpha = \alpha, \nu\alpha = \alpha \} \\ \!\!\! &=& \!\!\! \{ \alpha \in Spin(8) \,| \, \nu\alpha = \alpha \}\\ \!\!\! &=& \!\!\! \{ \alpha \in Spin(7) \,| \, \pi\alpha = \alpha \}. \end{aligned}

\operatorname{Spin}(8) /\{(1,1,1),(-1,-1,1)\} \cong \operatorname{Spin}(8) /\{(1,1,1),(1,-1,-1)\},

Spin(8)/\{(1,1,1),(-1,-1,1)\} \cong Spin(8)/\{(1,1,1),(1,-1,-1) \},

S O(8) \cong S s(8)

SO(8) \cong Ss(8).

G_{2}=\{\alpha \in \operatorname{Spin}(8) \mid \nu \alpha=\alpha\} .

G_2 = \{ \alpha \in Spin(8) \, | \, \nu\alpha = \alpha \}.

(\alpha x)(\alpha y)=\kappa \alpha(x y), \quad x, y \in \mathfrak{C},

\displaylines{\hfill (\alpha x)(\alpha y) = \kappa\alpha(xy), \quad x,y \in \text{\es {C}}, \hfill\text{(i)}}

a(\alpha x)=(\alpha x) a \quad \text { for all } \quad x \in \mathfrak{C} .

a(\alpha x)=(\alpha x)a \quad \text{ for all }\;\; x \in \text{\es {C}}.

\nu^{2}(-1,-1,1)=(1,-1,-1),

\nu^2(-1,-1,1)=(1,-1,-1),

\mathfrak{J}=\left\{X \in M(3, \mathfrak{C}) \mid X^{*}=X\right\},

\text{\es {J}} = \{ X \in M(3, \text{\es {C}}) \, | \, X^* = X \},

X=X(\xi, x)=\left(\begin{array}{ccc} \xi_{1} & x_{3} & \bar{x}_{2} \\ \bar{x}_{3} & \xi_{2} & x_{1} \\ x_{2} & \bar{x}_{1} & \xi_{3} \end{array}\right), \quad \xi_{i} \in \boldsymbol{R}, x_{i} \in \mathfrak{C} .

X = X(\xi, x) = \pmatrix{\xi_1 & x_3 & \overline{x}_2 \cr \overline{x}_3 & \xi_2 & x_1 \cr x_2 & \overline{x}_1 & \xi_3}, \quad \xi_i \in \text{$R$}, x_i \in \text{\es {C}}.

X \circ Y=\frac{1}{2}(X Y+Y X)

X \circ Y = \frac{1}{2}(XY + YX).

\begin{gathered} \operatorname{tr}(X)=\xi_{1}+\xi_{2}+\xi_{3}, \quad X=X(\xi, x), \\ (X, Y)=\operatorname{tr}(X \circ Y), \quad \operatorname{tr}(X, Y, Z)=(X, Y \circ Z) . \end{gathered}

\begin{array}{c} \text\mathrm{{tr}}(X) = \xi_1 + \xi_2 + \xi_3, \quad X = X(\xi, x), \vspace{1mm}\\ (X, Y) = \text\mathrm{{tr}}(X \circ Y), \quad \text\mathrm{{tr}}(X, Y, Z) = (X, Y \circ Z). \end{array}

X \times Y=\frac{1}{2}(2 X \circ Y-\operatorname{tr}(X) Y-\operatorname{tr}(Y) X+(\operatorname{tr}(X) \operatorname{tr}(Y)-(X, Y)) E),

X \times Y = \frac{1}{2}(2X \circ Y-\text\mathrm{{tr}}(X)Y - \text\mathrm{{tr}}(Y)X + (\text\mathrm{{tr}}(X)\text\mathrm{{tr}}(Y) - (X, Y))E),

(X, Y, Z)=(X, Y \times Z), \quad \operatorname{det} X=\frac{1}{3}(X, X, X) .

(X, Y, Z) = (X, Y \times Z), \quad \text\mathrm{{det}} X = \frac{1}{3}(X, X, X).

\begin{aligned} (X, Y) & =\sum_{i=1}^{3}\left(\xi_{i} \eta_{i}+2\left(x_{i}, y_{i}\right)\right), \\ \operatorname{tr}(X, Y, Z) & =\sum_{i=1}^{3}\left(\xi_{i} \eta_{i} \zeta_{i}+R\left(x_{i} y_{i+1} z_{i+2}+x_{i} z_{i+1} y_{i+2}\right)\right. \\ & +\xi_{i}\left(\left(y_{i+1}, z_{i+1}\right)+\left(y_{i+2}, z_{i+2}\right)\right)+\eta_{i}\left(\left(z_{i+1}, x_{i+1}\right)+\left(z_{i+2}, x_{i+2}\right)\right) \\ & \left.+\zeta_{i}\left(\left(x_{i+1}, y_{i+1}\right)+\left(x_{i+2}, y_{i+2}\right)\right)\right), \end{aligned}

\begin{aligned} (X, Y, Z) & =\sum_{i=1}^{3}\left(\frac{1}{2}\left(\xi_{i} \eta_{i+1} \zeta_{i+2}+\xi_{i} \eta_{i+2} \zeta_{i+1}\right)+R\left(x_{i} y_{i+1} z_{i+2}+x_{i} z_{i+1} y_{i+2}\right)\right. \\ & \left.-\left(\xi_{i}\left(y_{i}, z_{i}\right)+\eta_{i}\left(z_{i}, x_{i}\right)+\zeta_{i}\left(x_{i}, y_{i}\right)\right)\right) \\ \operatorname{det} X & =\xi_{1} \xi_{2} \xi_{3}+2 R\left(x_{1} x_{2} x_{3}\right)-\xi_{1} x_{1} \bar{x}_{1}-\xi_{2} x_{2} \bar{x}_{2}-\xi_{3} x_{3} \bar{x}_{3} \end{aligned}

X \times X=\left(\begin{array}{ccc} \xi_{2} \xi_{3}-x_{1} \bar{x}_{1} & \overline{x_{1} x_{2}}-\xi_{3} x_{3} & x_{3} x_{1}-\xi_{2} \bar{x}_{2} \\ x_{1} x_{2}-\xi_{3} \bar{x}_{3} & \xi_{3} \xi_{1}-x_{2} \bar{x}_{2} & \overline{x_{2} x_{3}}-\xi_{1} x_{1} \\ \overline{x_{3} x_{1}}-\xi_{2} x_{2} & x_{2} x_{3}-\xi_{1} \bar{x}_{1} & \xi_{1} \xi_{2}-x_{3} \bar{x}_{3} \end{array}\right), \quad X=X(\xi, x),

\begin{gathered} E_{1}=\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right), \quad E_{2}=\left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right), \quad E_{3}=\left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right), \\ F_{1}(x)=\left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & x \\ 0 & \bar{x} & 0 \end{array}\right), \quad F_{2}(x)=\left(\begin{array}{ccc} 0 & 0 & \bar{x} \\ 0 & 0 & 0 \\ x & 0 & 0 \end{array}\right), \quad F_{3}(x)=\left(\begin{array}{ccc} 0 & x & 0 \\ \bar{x} & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) . \end{gathered}

\left\{\begin{array} { l } { E _ { i } \circ E _ { i } = E _ { i } } \\ { E _ { i } \circ F _ { i } ( x ) = 0 } \\ { F _ { i } ( x ) \circ F _ { i } ( y ) = ( x , y ) ( E _ { i + 1 } + E _ { i + 2 } ) , } \end{array} \left\{\begin{array}{l} E_{i} \circ E_{j}=0, \quad i \neq j \\ E_{i} \circ F_{j}(x)=\frac{1}{2} F_{j}(x), \quad i \neq j \\ F_{i}(x) \circ F_{i+1}(y)=\frac{1}{2} F_{i+2}(\overline{x y}), \end{array}\right.\right.

\left\{\begin{array} { l } { E _ { i } \times E _ { i } = 0 } \\ { E _ { i } \times F _ { i } ( x ) = - \frac { 1 } { 2 } F _ { i } ( x ) } \\ { F _ { i } ( x ) \times F _ { i } ( y ) = - ( x , y ) E _ { i } , } \end{array} \left\{\begin{array}{l} E_{i} \times E_{i+1}=\frac{1}{2} E_{i+2} \\ E_{i} \times F_{j}(x)=0, \quad i \neq j \\ F_{i}(x) \times F_{i+1}(y)=\frac{1}{2} F_{i+2}(\overline{x y}), \end{array}\right.\right.

F_{4}=\left\{\alpha \in \operatorname{Iso}_{\boldsymbol{R}}(\mathfrak{J}) \mid \alpha(X \circ Y)=\alpha X \circ \alpha Y\right\} .

F_4 = \{ \alpha \in \text\mathrm{{Iso}}_{\text{${R}$}}(\text{\es {J}}) \, | \ \alpha(X \circ Y) = \alpha X \circ \alpha Y \}.

X \circ(X \circ X)-\operatorname{tr}(X) X^{2}+\frac{1}{2}\left(\operatorname{tr}(X)^{2}-\operatorname{tr}\left(X^{2}\right)\right) X=(\operatorname{det} X) E .

\displaylines{\hfill X \circ (X \circ X) - \text\mathrm{{tr}}(X)X^2 + \frac{1}{2}(\text\mathrm{{tr}}(X)^2 - \text\mathrm{{tr}}(X^2))X = (\text\mathrm{{det}} X)E. \hfill\text{(i)}}

X \circ(X \circ X)-\operatorname{tr}(\alpha X) X^{2}+\frac{1}{2}\left(\operatorname{tr}(\alpha X)^{2}-\operatorname{tr}\left(\left(\alpha X^{2}\right)\right) X=(\operatorname{det} \alpha X) E .\right.

\displaylines{\hfill X \circ (X \circ X) - \text\mathrm{{tr}}(\alpha X)X^2 + \frac{1}{2}(\text\mathrm{{tr}}(\alpha X)^2 - \text\mathrm{{tr}}((\alpha X^2))X = (\text\mathrm{{det}}\alpha X)E. \hfill\text{(ii)}}

\begin{aligned} (\operatorname{tr}(\alpha X) & -\operatorname{tr}(X)) X^{2}+\frac{1}{2}\left(\operatorname{tr}(X)^{2}-\operatorname{tr}(\alpha X)^{2}+\operatorname{tr}\left((\alpha X)^{2}\right)-\operatorname{tr}\left(X^{2}\right)\right) X \\ & =(\operatorname{det} X-\operatorname{det}(\alpha X)) E \end{aligned}

\begin{array}{l} (\text\mathrm{{tr}}(\alpha X) - \text\mathrm{{tr}}(X))X^2 + {\frac{1}{2}}(\text\mathrm{{tr}}(X)^2 - \text\mathrm{{tr}}(\alpha X)^2 + \text\mathrm{{tr}}((\alpha X)^2)-\text\mathrm{{tr}} (X^2))X \vspace{1mm}\\ \qquad \qquad = (\text\mathrm{{det}} X - \text\mathrm{{det}}(\alpha X))E. \end{array}

\begin{aligned} & \operatorname{tr}\left(\alpha F_{i}\left(e_{j}\right)\right)\left(E_{i+1}+E_{i+2}\right)+\frac{1}{2}\left(-\operatorname{tr}\left(\alpha F_{i}\left(e_{j}\right)\right)^{2}+\operatorname{tr}\left(\left(\alpha F_{i}\left(e_{j}\right)\right)^{2}\right)-2\right) F_{i}\left(e_{j}\right) \\ & \quad=-\operatorname{det}\left(\alpha F_{i}\left(e_{j}\right)\right) E . \end{aligned}

\begin{array}{l} \text\mathrm{{tr}}(\alpha F_i(e_j))(E_{i+1} + E_{i+2}) + {\frac{1}{2}}(-\text\mathrm{{tr}}(\alpha F_i(e_j))^2 + \text\mathrm{{tr}}((\alpha F_i(e_j))^2) - 2)F_i(e_j) \vspace{1mm}\\ \qquad \qquad = -\text\mathrm{{det}}(\alpha F_i(e_j))E. \end{array}

\operatorname{tr}\left(\alpha F_{i}\left(e_{j}\right)\right)=0\left(=\operatorname{tr}\left(F_{i}\left(e_{j}\right)\right)\right)

\text\mathrm{{tr}}(\alpha F_i(e_j)) = 0 \, \, (= \text\mathrm{{tr}}(F_i(e_j)))

\operatorname{tr}\left(\alpha E_{i}\right)=\operatorname{tr}\left(\alpha\left(E-F_{i}(1)^{2}\right)\right)=\operatorname{tr}(E)-\operatorname{tr}\left(\left(\alpha F_{i}(1)\right)^{2}\right)=3-2=1=\operatorname{tr}\left(E_{i}\right),

\text\mathrm{{tr}}(\alpha E_i) = \text\mathrm{{tr}}(\alpha(E - F_i(1)^2)) = \text\mathrm{{tr}}(E) - \text\mathrm{{tr}}((\alpha F_i(1))^2) = 3 - 2 = 1 = \text\mathrm{{tr}}(E_i),

\operatorname{det}\left({ }^{t} \alpha^{-1} X\right)=\operatorname{det}\left({ }^{t} \alpha X\right)=\operatorname{det} X, \quad \text { for all } X \in \mathfrak{J} .

\text\mathrm{{det}}\,({}^t\alpha^{-1}X) = \text\mathrm{{det}}\,({}^t\alpha X) = \text\mathrm{{det}}\,X, \quad \text{\mathit{for all}} \, \, \, X \in \text{\es {J}}.

\begin{aligned} { }^{t} \alpha^{-1}(Y \times & Y) \times{ }^{t} \alpha^{-1}(Y \times Y)=(\alpha Y \times \alpha Y) \times(\alpha Y \times \alpha Y) \quad(\text { Lemma 2.2.2.(4) }) \\ & =(\operatorname{det} \alpha Y) \alpha Y \quad(\text { Lemma 2.1.1.(3)) }=(\operatorname{det} Y) \alpha Y=\alpha((\operatorname{det} Y) Y) \\ & =\alpha((Y \times Y) \times(Y \times Y)), \quad Y \in \mathfrak{J} . \end{aligned}

{ }^{t} \alpha^{-1}((\operatorname{det} X) X) \times{ }^{t} \alpha^{-1}((\operatorname{det} X) X)=\alpha((\operatorname{det} X) X \times(\operatorname{det} X) X) .

{}^t\alpha^{-1}((\text\mathrm{{det}}\,X)X) \times {}^t\alpha^{-1}((\text\mathrm{{det}}\,X)X) = \alpha((\text\mathrm{{det}}\,X)X \times (\text\mathrm{{det}}\,X)X).

\begin{aligned} & 3 \operatorname{det}\left({ }^{t} \alpha^{-1} X\right)=\left({ }^{t} \alpha^{-1} X,{ }^{t} \alpha^{-1} X \times{ }^{t} \alpha^{-1} X\right) \\ & \quad=\left({ }^{t} \alpha^{-1} X, \alpha(X \times X)\right)=(X, X \times X)=3 \operatorname{det} X, \end{aligned}

\begin{array}{l} 3\text\mathrm{{det}}\,({}^t\alpha^{-1}X) = ({}^t \alpha^{-1}X,{}^t \alpha^{-1}X \times {}^t \alpha^{-1} X) \vspace{1mm}\\ \hspace{15mm} = ({}^t\alpha^{-1}X, \alpha(X \times X)) = (X, X \times X) = 3\text\mathrm{{det}}\,X, \end{array}

\alpha(X \circ Y)=\alpha X \circ \alpha Y .

\operatorname{tr}(\alpha X, \alpha Y, \alpha Z)=\operatorname{tr}(X, Y, Z),(\alpha X, \alpha Y)=(X, Y) .

\operatorname{det}(\alpha X)=\operatorname{det} X,(\alpha X, \alpha Y)=(X, Y) .

\operatorname{det}(\alpha X)=\operatorname{det} X, \alpha E=E .

\alpha(X \times Y)=\alpha X \times \alpha Y .

\begin{aligned} & 3 \operatorname{det}(\alpha X)=\operatorname{tr}(\alpha X, \alpha X, \alpha X)-\frac{3}{2} \operatorname{tr}(\alpha X)(\alpha X, \alpha X)+\frac{1}{2} \operatorname{tr}(\alpha X)^{3} \\ & \quad=\operatorname{tr}(X, X, X)-\frac{3}{2} \operatorname{tr}(X)(X, X)+\frac{1}{2} \operatorname{tr}(X)^{3}=3 \operatorname{det} X . \end{aligned}

\begin{array}{l} 3\text\mathrm{{det}}(\alpha X) = \text\mathrm{{tr}}(\alpha X, \alpha X, \alpha X) - \frac{3}{2}\text\mathrm{{tr}}(\alpha X)(\alpha X, \alpha X) + \frac{1}{2}\text\mathrm{{tr}}(\alpha X)^3 \vspace{1mm}\\ \quad \quad = \text\mathrm{{tr}}(X, X, X) - \frac{3}{2}\text\mathrm{{tr}}(X)(X, X) + \frac{1}{2}\text\mathrm{{tr}}(X)^3 = 3\text\mathrm{{det}} X. \end{array}

\begin{aligned} & (\operatorname{det}(\alpha X)) \alpha X=(\alpha X \times \alpha X) \times(\alpha X \times \alpha X) \quad(\text { Lemma 2.1.1. } \\ & =\alpha((X \times X) \times(X \times X))=(\operatorname{det} X) \alpha X \quad(\text { Lemma 2.1.1. } \end{aligned}

\alpha X \circ \alpha E=\alpha(X \times E)=\frac{1}{2} \alpha(\operatorname{tr}(X) E-X),

\alpha X \circ \alpha E = \alpha(X \times E) = \frac{1}{2}\alpha(\text\mathrm{{tr}}(X)E -X),

\alpha X \times P=\frac{1}{2} \operatorname{tr}(X) P-\frac{1}{2} \alpha X, \quad X \in \mathfrak{J} .

\alpha X \times P = \frac{1}{2}\text\mathrm{{tr}}(X)P - \frac{1}{2}\alpha X, \quad X \in \text{\es {J}}.

\begin{aligned} & \frac{1}{2}\left(\rho_{2} E_{3}+\rho_{3} E_{2}-F_{1}\left(p_{1}\right)\right) \\ & \quad=\frac{1}{2}\left(\lambda\left(\rho_{1} E_{1}+\rho_{2} E_{2}+\rho_{3} E_{3}+F_{1}\left(p_{1}\right)+F_{2}\left(p_{2}\right)+F_{3}\left(p_{3}\right)\right)-E_{1}\right) \end{aligned}

\begin{array}{l} \frac{1}{2}(\rho_2E_3 + \rho_3E_2 - F_1(p_1)) \\ \qquad = \frac{1}{2}(\lambda(\rho_1E_1 + \rho_2E_2 + \rho_3E_3 + F_1(p_1) + F_2(p_2) + F_3(p_3)) - E_1), \end{array}

0=\lambda \rho_{1}-1, \quad \rho_{3}=\lambda \rho_{2}, \quad \rho_{2}=\lambda \rho_{3}, \quad-p_{1}=\lambda p_{1}, \quad 0=\lambda p_{2}, \quad 0=\lambda p_{3} .

0 = \lambda\rho_1 - 1, \quad \rho_3 = \lambda\rho_2, \quad \rho_2 = \lambda\rho_3, \quad - p_1 = \lambda p_1, \quad 0 = \lambda p_2, \quad 0 = \lambda p_3.

-\frac{1}{2} \rho_{1} F_{1}(1)=\frac{1}{2}\left(\mu\left(\rho_{1} E_{1}+\rho_{2} E_{2}+\rho_{3} E_{3}\right)-F_{1}(1)\right),

- \frac{1}{2}\rho_1F_1(1) = \frac{1}{2}(\mu(\rho_1E_1 + \rho_2E_2 + \rho_3E_3) - F_1(1)),

\begin{aligned} & (4) \Rightarrow(2) \operatorname{tr}(\alpha X)=(\alpha X, E, E)=(\alpha X, \alpha E, \alpha E)=(X, E, E) \text { (Lemma 2.2.2.(2)) } \\ = & \operatorname{tr}(X) \text {. Hence } \frac{1}{2}(\operatorname{tr}(X) \operatorname{tr}(Y)-(X, Y))=(X, Y, E) \text { (Lemma 2.1.1) }=(\alpha X, \alpha Y, \alpha E) \\ = & (\alpha X, \alpha Y, E)=\frac{1}{2}(\operatorname{tr}(\alpha X) \operatorname{tr}(\alpha Y)-(\alpha X, \alpha Y))=\frac{1}{2}(\operatorname{tr}(X) \operatorname{tr}(Y)-(\alpha X, \alpha Y)) \text {. There- } \end{aligned}

(\alpha X, \alpha Y)=(X, Y) .

(\alpha X, \alpha Y) = (X, Y).

\operatorname{tr}(\alpha X, \alpha Y, \alpha Z)=\operatorname{tr}(X, Y, Z) .

\text\mathrm{{tr}}(\alpha X, \alpha Y, \alpha Z) = \text\mathrm{{tr}}(X, Y, Z).

O(27)=O(\mathfrak{J})=\left\{\alpha \in \operatorname{Iso}_{\boldsymbol{R}}(\mathfrak{J}) \mid(\alpha X, \alpha Y)=(X, Y)\right\} \quad \text { (Lemma 2.2.4.(2)). }

O(27) = O(\text{\es {J}}) = \{ \alpha \in \text\mathrm{{Iso}}_{\text{${R}$}}(\text{\es {J}}) \, | \, (\alpha X, \alpha Y) = (X, Y) \}\;\;\text{(Lemma 2.2.4.(2))}.

\widetilde{\alpha}\left(\begin{array}{ccc} \xi_{1} & x_{3} & \bar{x}_{2} \\ \bar{x}_{3} & \xi_{2} & x_{1} \\ x_{2} & \bar{x}_{1} & \xi_{3} \end{array}\right)=\left(\begin{array}{ccc} \xi_{1} & \alpha x_{3} & \overline{\alpha x_{2}} \\ \overline{\alpha x_{3}} & \xi_{2} & \alpha x_{1} \\ \alpha x_{2} & \overline{\alpha x_{1}} & \xi_{3} \end{array}\right) .

\begin{aligned} E_{6(-26)} & =\left\{\alpha \in \operatorname{Iso}_{\boldsymbol{R}}(\mathfrak{J}) \mid \operatorname{det}(\alpha X)=\operatorname{det} X\right\} \\ & =\left\{\left.\alpha \in \operatorname{Iso}_{\boldsymbol{R}}(\mathfrak{J})\right|^{t} \alpha^{-1}(X \times Y)=\alpha X \times \alpha Y\right\} . \end{aligned}

\begin{aligned} E_{6({}-26)} \!\!\! &=& \!\!\! \{\alpha \in \text\mathrm{{Iso}}_{\text{${R}$}}(\text{\es {J}}) \, | \, \text\mathrm{{det}}\,(\alpha X) = \text\mathrm{{det}}\,X \} \\ \!\!\! &=&\!\!\! \{\alpha \in \text\mathrm{{Iso}}_{\text{${R}$}}(\text{\es {J}}) \, | \, {}^t\alpha^{-1}(X \times Y) = \alpha X \times \alpha Y \}. \end{aligned}

\begin{aligned} \mathfrak{e}_{6(-26)} & =\left\{\phi \in \operatorname{Hom}_{\boldsymbol{R}}(\mathfrak{J}) \mid(\phi X, X, X)=0\right\} \\ & =\left\{\phi \in \operatorname{Hom}_{\boldsymbol{R}}(\mathfrak{J}) \mid(\phi X, Y, Z)+(X, \phi Y, Z)+(X, Y, \phi Z)=0\right\} \\ & =\left\{\phi \in \operatorname{Hom}_{\boldsymbol{R}}(\mathfrak{J}) \mid-{ }^{t} \phi(X \times Y)=\phi X \times Y+X \times \phi Y\right\} . \end{aligned}

\begin{aligned} \text{\es {e}}_{6(-26)} \!\!\! &=& \!\!\! \{ \phi \in \text\mathrm{{Hom}}_{\text{${R}$}}(\text{\es {J}}) \, | \, (\phi X, X, X) = 0 \} \\ \!\!\! &=& \!\!\! \{ \phi \in \text\mathrm{{Hom}}_{\text{${R}$}}(\text{\es {J}}) \, | \, (\phi X, Y, Z) + (X, \phi Y, Z) + (X, Y, \phi Z) = 0 \} \\ \!\!\! &=& \!\!\! \{ \phi \in \text\mathrm{{Hom}}_{\text{${R}$}}(\text{\es {J}}) \, | \, -{}^t\phi(X \times Y) = \phi X \times Y + X \times \phi Y \}. \end{aligned}

\begin{aligned} \mathfrak{f}_{4} & =\left\{\delta \in \operatorname{Hom}_{\boldsymbol{R}}(\mathfrak{J}) \mid \delta(X \circ Y)=\delta X \circ Y+X \circ \delta Y\right\} \\ & =\left\{\delta \in \operatorname{Hom}_{\boldsymbol{R}}(\mathfrak{J}) \left\lvert\, \begin{array}{c} \operatorname{tr}(\delta X, Y, Z)+\operatorname{tr}(X, \delta Y, Z)+\operatorname{tr}(X, Y, \delta Z)=0 \\ (\delta X, Y)+(X, \delta Y)=0 \end{array}\right.\right\} \\ & =\left\{\delta \in \operatorname{Hom}_{\boldsymbol{R}}(\mathfrak{J}) \mid(\delta X, X, X)=0,(\delta X, Y)+(X, \delta Y)=0\right\} \\ & =\left\{\delta \in \operatorname{Hom}_{\boldsymbol{R}}(\mathfrak{J}) \mid(\delta X, X, X)=0, \delta E=0\right\} \\ & =\left\{\delta \in \operatorname{Hom}_{\boldsymbol{R}}(\mathfrak{J}) \mid \delta(X \times Y)=\delta X \times Y+X \times \delta Y\right\} . \end{aligned}

\mathfrak{M}^{-}=\left\{A \in M(3, \mathfrak{C}) \mid A^{*}=-A\right\} .

\text{\es {M}}^- = \{ A \in M(3, \text{\es {C}}) \, | \, A^* = -A \}.

[X, Y]=X Y-Y X .

[X, Y] = XY - YX.

\widetilde{A} X=\frac{1}{2}[A, X], \quad X \in \mathfrak{J} .

\widetilde{A}X = \frac{1}{2}[A, X], \quad X \in \text{\es {J}}.

[X, X X]=a E .

[X, XX] = aE.

a_{i j}=\sum_{k, l}\left(x_{i k}\left(x_{k l} x_{l j}\right)-\left(x_{i k} x_{k l}\right) x_{l j}\right)=-\sum_{k, l}\left\{x_{i k}, x_{k l}, x_{l j}\right\}, \quad i, j=1,2,3 .

a_{ij} = \sum_{k,l}(x_{ik}(x_{kl}x_{lj}) - (x_{ik}x_{kl})x_{lj}) = - \sum_{k,l}\{x_{ik}, x_{kl}, x_{lj}\}, \quad i, j = 1, 2, 3.

\begin{aligned} a_{11} & =-\left\{x_{12}, x_{23}, x_{31}\right\}-\left\{x_{13}, x_{32}, x_{21}\right\}, \\ a_{22} & =-\left\{x_{21}, x_{13}, x_{32}\right\}-\left\{x_{23}, x_{31}, x_{12}\right\}, \\ a_{33} & =-\left\{x_{31}, x_{12}, x_{23}\right\}-\left\{x_{32}, x_{21}, x_{13}\right\}, \end{aligned}

\begin{aligned} a_{11} \!\!\! &=& \!\!\! - \{x_{12}, x_{23}, x_{31}\} - \{x_{13}, x_{32}, x_{21}\}, \vspace{1mm}\\ a_{22} \!\!\! &=& \!\!\! - \{x_{21}, x_{13}, x_{32}\} - \{x_{23}, x_{31}, x_{12}\}, \vspace{1mm}\\ a_{33} \!\!\! &=& \!\!\! - \{x_{31}, x_{12}, x_{23}\} - \{x_{32}, x_{21}, x_{13}\}, \end{aligned}

(X, Y)=\frac{1}{2} \operatorname{tr}\left(X Y+Y^{*} X^{*}\right) .

(X, Y) = \frac{1}{2}\text\mathrm{{tr}}(XY + Y^*X^*).

(X Y, Z)=(Y Z, X)=(Z X, Y)=(X, Y Z)=(Y, Z X)=(Z, X Y) .

(XY, Z) = (YZ, X) = (ZX, Y) = (X, YZ) = (Y, ZX) = (Z, XY).

\begin{aligned} (X Y, Z) & =R(X Y, Z)=\frac{1}{2} R\left(\operatorname{tr}\left((X Y) Z+Z^{*}\left(Y^{*} X^{*}\right)\right)\right) \\ & =\frac{1}{2} R\left(\sum_{i, j, k}\left(\left(x_{i j} y_{j k}\right) z_{k i}+\bar{z}_{j i}\left(\bar{y}_{k j} \bar{x}_{i k}\right)\right)\right) \\ & =\frac{1}{2} R\left(\sum_{i, j, k}\left(\left(y_{j k} z_{k i}\right) x_{i j}+\bar{x}_{i k}\left(\bar{z}_{j i} \bar{y}_{k j}\right)\right)\right) \\ & =\frac{1}{2} R\left(\operatorname{tr}\left((Y Z) X+X^{*}\left(Z^{*} Y^{*}\right)\right)\right)=(Y Z, X) . \end{aligned}

\left\{\begin{array}{l} ([A, X], Y)+(X,[A, Y])=0, \quad X, Y \in \mathfrak{J} \\ \operatorname{tr}([A, X], Y, Z)+\operatorname{tr}(X,[A, Y], Z)+\operatorname{tr}(X, Y,[A, Z])=0, \quad X, Y, Z \in \mathfrak{J} \end{array}\right.

\left\{ \begin{array}{l} ([A, X], Y) + (X, [A, Y]) = 0, \quad X, Y \in \text{\es {J}}, \vspace{1mm}\\ \text\mathrm{{tr}}([A, X], Y, Z) + \text\mathrm{{tr}}(X, [A, Y], Z) + \text\mathrm{{tr}}(X, Y, [A, Z]) = 0, \quad X, Y, Z \in \text{\es {J}}. \end{array}\right.

([A, X], X X)=0, \quad X \in \mathfrak{J} .

([A, X], XX) = 0, \quad X \in \text{\es {J}}.

\begin{aligned} ([A, X], X X) & =(A X, X X)-(X A, X X) \\ & =(A, X(X X))-(A,(X X) X) \quad(\text { Lemma 2.3.5 }) \\ & =(A,[X, X X])=(A, a E)=\frac{1}{2} \operatorname{tr}\left(A a+\bar{a} A^{*}\right) \\ & =\frac{1}{2} \operatorname{tr}(A a-a A)=\frac{1}{2}(\operatorname{tr}(A) a-a \operatorname{tr}(A))=0 . \end{aligned}

([A, X], Y Z+Z Y)+([A, Y], X Z+Z X)+([A, Z], X Y+Y X)=0,

([A, X],YZ + ZY) + ([A, Y], XZ + ZX) + ([A, Z], XY + YX) = 0,

A_{1}(a)=\left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & a \\ 0 & -\bar{a} & 0 \end{array}\right), \quad A_{2}(a)=\left(\begin{array}{ccc} 0 & 0 & -\bar{a} \\ 0 & 0 & 0 \\ a & 0 & 0 \end{array}\right), \quad A_{3}(a)=\left(\begin{array}{ccc} 0 & a & 0 \\ -\bar{a} & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) .

\left\{\begin{array} { l } { \widetilde { A } _ { i } ( a ) E _ { i } = 0 } \\ { \widetilde { A } _ { i } ( a ) E _ { i + 1 } = - \frac { 1 } { 2 } F _ { i } ( a ) } \\ { \widetilde { A } _ { i } ( a ) E _ { i + 2 } = \frac { 1 } { 2 } F _ { i } ( a ) , } \end{array} \quad \left\{\begin{array}{l} \widetilde{A}_{i}(a) F_{i}(x)=(a, x)\left(E_{i+1}-E_{i+2}\right) \\ \widetilde{A}_{i}(a) F_{i+1}(x)=\frac{1}{2} F_{i+2}(\overline{a x}) \\ \widetilde{A}_{i}(a) F_{i+2}(x)=-\frac{1}{2} F_{i+1}(\overline{x a}) . \end{array}\right.\right.

\mathfrak{d}_{4}=\left\{\delta \in \mathfrak{f}_{4} \mid \delta E_{i}=0, i=1,2,3\right\}

\text{\es {d}}_4 = \{ \delta \in \text{\es {f}}_4 \, | \, \delta E_i = 0, i = 1, 2, 3 \}

\mathfrak{D}_{4}=\left\{D \in \operatorname{Hom}_{\boldsymbol{R}}(\mathfrak{C}) \mid(D x, y)+(x, D y)=0\right\}

\text{\es {D}}_4 = \{ D \in \text\mathrm{{Hom}}_{\text{${R}$}}(\text{\es {C}}) \, | \, (Dx, y) + (x, Dy) = 0\}

\mathfrak{D}_{4} \ni D_{1} \longrightarrow \delta \in \mathfrak{d}_{4}

\text{\es {D}}_4 \ni D_1 \longrightarrow \delta \in \text{\es {d}}_4

\delta\left(\begin{array}{ccc} \xi_{1} & x_{3} & \bar{x}_{2} \\ \bar{x}_{3} & \xi_{2} & x_{1} \\ x_{2} & \bar{x}_{1} & \xi_{3} \end{array}\right)=\left(\begin{array}{ccc} 0 & D_{3} x_{3} & \overline{D_{2} x_{2}} \\ \overline{D_{3} x_{3}} & 0 & D_{1} x_{1} \\ D_{2} x_{2} & \overline{D_{1} x_{1}} & 0 \end{array}\right),

\delta\pmatrix{\xi_1 & x_3 & \overline{x}_2 \cr \overline{x}_3 & \xi_2 & x_1 \cr x_2 & \overline{x}_1 & \xi_3} = \pmatrix{0 & D_3x_3 & \overline{D_2x_2} \vspace{0.5mm}\cr \overline{D_3x_3} & 0 & D_1x_1 \vspace{0.5mm}\cr D_2x_2 & \overline{D_1x_1} & 0},

\left(D_{1} x\right) y+x\left(D_{2} y\right)=\overline{D_{3}(\overline{x y})}, \quad x, y \in \mathfrak{C} .

(D_1x)y + x(D_2y) = \overline{D_3(\overline{xy})}, \quad x, y \in \text{\es {C}}.

\begin{aligned} & (\delta X, X, X)=(\delta X, X \times X) \\ & \quad=\left(\left(\begin{array}{ccc} 0 & D_{3} x_{3} & \overline{D_{2} x_{2}} \\ \overline{D_{3} x_{3}} & 0 & D_{1} x_{1} \\ D_{2} x_{2} & \overline{D_{1} x_{1}} & 0 \end{array}\right),\left(\begin{array}{ccc} \xi_{2} \xi_{3}-x_{1} \overline{x_{1}} & \overline{x_{1} x_{2}}-\xi_{3} x_{3} & x_{3} x_{1}-\xi_{2} \bar{x}_{2} \\ x_{1} x_{2}-\xi_{3} \bar{x}_{3} & \xi_{3} \xi_{1}-x_{2} \bar{x}_{2} & \overline{x_{2} x_{3}}-\xi_{1} x_{1} \\ \overline{x_{3} x_{1}}-\xi_{2} x_{2} & x_{2} x_{3}-\xi_{1} \bar{x}_{1} & \xi_{1} \xi_{2}-x_{3} \bar{x}_{3} \end{array}\right)\right) \\ & \quad=2\left(D_{1} x_{1}, \overline{x_{2} x_{3}}-\xi_{1} x_{1}\right)+2\left(D_{2} x_{2}, \overline{x_{3} x_{1}}-\xi_{2} x_{2}\right)+2\left(D_{3} x_{3}, \overline{x_{1} x_{2}}-\xi_{3} x_{3}\right) \\ & \quad=2\left(\left(D_{1} x_{1}, \overline{x_{2} x_{3}}\right)+\left(D_{2} x_{2}, \overline{x_{3} x_{1}}\right)+\left(D_{3} x_{3}, \overline{x_{1} x_{2}}\right)\right) \\ & \quad=2\left(\left(D_{1} x_{1}, \overline{x_{2} x_{3}}\right)+\left(\overline{D_{2} x_{2}}, x_{3} x_{1}\right)+\left(\overline{D_{3} x_{3}}, x_{1} x_{2}\right)\right) \end{aligned}

\begin{aligned} & \left(D_{1} x_{1}, \overline{x_{2} x_{3}}\right)=-\left(x_{1}, D_{1}\left(\overline{x_{2} x_{3}}\right)\right)=-\left(x_{1}, \overline{\left(D_{2} x_{2}\right) x_{3}+x_{2}\left(D_{3} x_{3}\right)}\right) \\ & \quad=-\left(x_{1}, \overline{x_{3}} \overline{\left(D_{2} x_{2}\right)}+\overline{\left(D_{3} x_{3}\right)} \overline{x_{2}}\right)=-\left(x_{3} x_{1}, \overline{D_{2} x_{2}}\right)-\left(x_{1} x_{2}, \overline{D_{3} x_{3}}\right) . \end{aligned}

\begin{array}{l} (D_1x_1, \overline{x_2x_3}) = - (x_1, D_1(\overline{x_2x_3})) = - (x_1, \overline{(D_2x_2)x_3 + x_2(D_3x_3)}) \vspace{1.5mm}\\ \qquad = - (x_1, \overline{x_3}\overline{(D_2x_2)} + \overline{(D_3x_3)}\overline{x_2}) = - (x_3x_1, \overline{D_2x_2}) - (x_1x_2, \overline{D_3x_3}). \end{array}

\mathfrak{J}_{i}=\left\{F_{i}(x) \mid x \in \mathfrak{C}\right\}=\left\{X \in \mathfrak{J} \mid 2 E_{i+1} \circ X=2 E_{i+2} \circ X=X\right\} .

\text{\es {J}}_i = \{F_i(x) \, | \, x \in \text{\es {C}} \} = \{X \in \text{\es {J}} \, | \, 2E_{i+1} \circ X = 2E_{i+2} \circ X = X \}.

\delta F_{i}(x)=F_{i}\left(D_{i} x\right), \quad x \in \mathfrak{C},

\delta F_i(x) = F_i(D_ix), \quad x \in \text{\es {C}},

\left(D_{i} x, y\right)+\left(x, D_{i} y\right)=0, \quad x, y \in \mathfrak{C} .

(D_ix, y) + (x, D_iy) = 0, \quad x, y \in \text{\es {C}}.

\left(D_{1} x\right) y+x\left(D_{2} y\right)=\overline{D_{3}(\overline{x y})}, \quad x, y \in \mathfrak{C} .

(D_1x)y + x(D_2y) = \overline{D_3(\overline{xy})}, \quad x, y \in \text{\es {C}}.

\delta=D+\widetilde{A}, \quad D \in \mathfrak{d}_{4}, A \in \mathfrak{M}^{-}, \operatorname{diag} A=0,

\delta = D + \widetilde{A}, \quad D \in \text{\es {d}}_4, A \in \text{\es {M}}^-, \text\mathrm{{diag}} A = 0,

\operatorname{dim} \mathfrak{f}_{4}=28+24=52 .

\dim\text{\es {f}}_4 = 28 + 24 = 52.

2 \delta E_{i} \circ E_{i}=\delta E_{i}, \quad \delta E_{i} \circ E_{j}+E_{i} \circ \delta E_{j}=0 .

2\delta E_i \circ E_i = \delta E_i, \quad \delta E_i \circ E_j + E_i \circ \delta E_j = 0.

\delta E_{1}=\left(\begin{array}{ccc} 0 & -a_{3} & \bar{a}_{2} \\ -\bar{a}_{3} & 0 & 0 \\ a_{2} & 0 & 0 \end{array}\right), \delta E_{2}=\left(\begin{array}{ccc} 0 & a_{3} & 0 \\ \bar{a}_{3} & 0 & -a_{1} \\ 0 & -\bar{a}_{1} & 0 \end{array}\right), \delta E_{3}=\left(\begin{array}{ccc} 0 & 0 & -\bar{a}_{2} \\ 0 & 0 & a_{1} \\ -a_{2} & \bar{a}_{1} & 0 \end{array}\right),

\delta E_1 = \pmatrix{0 \! & \! -a_3 \! & \! \overline{a}_2 \cr -\overline{a}_3 \! & \! 0 \! & \! 0 \cr a_2 \! & \! 0 \! & \! 0}, \delta E_2 = \pmatrix{0 \! & \! a_3 \! & \! 0 \cr \overline{a}_3 \! & \! 0 \! & \! -a_1 \cr 0 \! & \! -\overline{a}_1 \! & \! 0}, \delta E_3 = \pmatrix{0 \! & \! 0 \! & \! -\overline{a}_2 \cr 0 \! & \! 0 \! & \! a_1 \cr -a_2 \! & \! \overline{a}_1 \! & \! 0},

\delta E_{i}=\widetilde{A} E_{i}, \quad i=1,2,3 .

\delta E_i = \widetilde{A}E_i, \quad i = 1, 2, 3.

D+\widetilde{A}=0, \quad D \in \mathfrak{d}_{4}, A \in \mathfrak{M}^{-}, \operatorname{diag} A=0, \text { then } D=0, A=0 .

D + \widetilde{A} = 0, \; \; D \in \text{\es {d}}_4, A \in \text{\es {M}}^-, \text\mathrm{{diag}} A = 0, \;\; \text{then} \;\; D = 0, A = 0.

\overline{X_{1}+i X_{2}}=\bar{X}_{1}+i \bar{X}_{2}, \quad \tau\left(X_{1}+i X_{2}\right)=X_{1}-i X_{2}, \quad X_{i} \in \mathfrak{J} .

\overline{X_1 + i X_2} = \overline{X}_1 + i\overline{X}_2,\;\; \tau(X_1 + iX_2) = X_1 - iX_2, \quad X_i \in \text{\es {J}}.

\tau(X \circ Y)=\tau X \circ \tau Y, \quad \tau(X \times Y)=\tau X \times \tau Y, \quad X, Y \in \mathfrak{J}^{C} .

\tau(X \circ Y) = \tau X \circ \tau Y,\;\; \tau(X \times Y) = \tau X \times \tau Y, \quad X, Y \in \text{\es {J}}^C.

\begin{aligned} \mathfrak{e}_{6}^{C} & =\left\{\phi \in \operatorname{Hom}_{C}\left(\mathfrak{J}^{C}\right) \mid(\phi X, X, X)=0\right\} \\ & =\left\{\phi \in \operatorname{Hom}_{C}\left(\mathfrak{J}^{C}\right) \mid-{ }^{t} \phi(X \times Y)=\phi X \times Y+X \times \phi Y\right\}, \\ \mathfrak{f}_{4}^{C} & =\left\{\delta \in \operatorname{Hom}_{C}\left(\mathfrak{J}^{C}\right) \mid \delta(X \circ Y)=\delta X \circ Y+X \circ \delta Y\right\} \\ & =\left\{\delta \in \operatorname{Hom}_{C}\left(\mathfrak{J}^{C}\right) \mid \delta(X \times Y)=\delta X \times Y+X \times \delta Y\right\} . \end{aligned}

\begin{aligned} {\text{\es {e}}_6}^C \!\!\!&=&\!\!\! \{\phi \in \text\mathrm{{Hom}}_C(\text{\es {J}}^C) \, | \, (\phi X, X, X) = 0\} \vspace{1mm}\\ \!\!\!&=&\!\!\! \{\phi \in \text\mathrm{{Hom}}_C(\text{\es {J}}^C) \, | \, -{}^t\phi(X \times Y) = \phi X \times Y + X \times \phi Y\}, \vspace{1mm}\\ {\text{\es {f}}_4}^C \!\!\!&=&\!\!\! \{\delta \in \text\mathrm{{Hom}}_C(\text{\es {J}}^C) \, | \, \delta(X \circ Y) = \delta X \circ Y + X \circ \delta Y\} \vspace{1mm}\\ \!\!\!&=&\!\!\! \{\delta \in \text\mathrm{{Hom}}_C(\text{\es {J}}^C) \, | \, \delta(X \times Y) = \delta X \times Y + X \times \delta Y\}. \end{aligned}

\widetilde{A} X=A \circ X, \quad X \in \mathfrak{J}^{C} .

\widetilde{A}X = A \circ X, \quad X \in \text{\es {J}}^C.

=(A,(\operatorname{det} X) E)=(\operatorname{det} X)(A, E)=(\operatorname{det} X) \operatorname{tr}(A)=0, \quad X \in \mathfrak{J}^{C} .

\begin{aligned} & {[\widetilde{A}, \widetilde{B}]=\left[\left(A-\frac{1}{3} \operatorname{tr}(A) E\right)^{\sim},\left(B-\frac{1}{3} \operatorname{tr}(B) E\right)^{\sim}\right] \in e_{6}^{C},} \\ & {[\widetilde{A}, \widetilde{B}] E=\widetilde{A}(\widetilde{B} E)-\widetilde{B}(\widetilde{A} E)=A \circ B-B \circ A=0 .} \end{aligned}

\begin{array}{ll} {\left[\widetilde{E}_{i}, \widetilde{E}_{j}\right]=0,} & {\left[\widetilde{E}_{i}, \widetilde{F}_{i}(a)\right]=0,} \\ {\left[\widetilde{E}_{i}, \widetilde{F}_{i+1}(a)\right]=-\frac{1}{2} \widetilde{A}_{i+1}(a),} & {\left[\widetilde{E}_{i}, \widetilde{F}_{i+2}(a)\right]=\frac{1}{2} \widetilde{A}_{i+2}(a),} \\ {\left[\widetilde{F}_{i}(a), \widetilde{F}_{i}(b)\right] \in \mathfrak{d}_{4}^{C},} & {\left[\widetilde{F}_{i}(a), \widetilde{F}_{i+1}(b)\right]=-\frac{1}{2} \widetilde{A}_{i+2}(\overline{a b}),} \\ {\left[D, \widetilde{A}_{i}(a)\right]=\widetilde{A}_{i}\left(D_{i} a\right),} & \text { where } D=\left(D_{1}, D_{2}, D_{3}\right) \in \mathfrak{d}_{4}^{C}, \\ {\left[\widetilde{A}_{i}(a), \widetilde{A}_{i}(b)\right] \in \mathfrak{d}_{4}^{C},} & {\left[\widetilde{A}_{i}(a), \widetilde{A}_{i+1}(b)\right]=-\frac{1}{2} \widetilde{A}_{i+2}(\overline{a b}),} \\ {\left[\widetilde{F}_{1}\left(e_{i}\right), \widetilde{F}_{1}\left(e_{j}\right)\right]=G_{i j} .} & {\left[\widetilde{A}_{i}(a), \widetilde{A}_{i+2}(b)\right]=\frac{1}{2} \widetilde{A}_{i+1}(\overline{b a}) .} \end{array}

x \xrightarrow{G_{i 0}} x_{i} e_{0}-x_{0} e_{i} \xrightarrow{G_{j 0}(j \neq i)} x_{i} e_{j} \xrightarrow{x_{j}{ }^{-1} G_{0 j}} e_{0}=1 .

x \stackrel{G_{i0}}{\longrightarrow}x_ie_0 - x_0e_i \stackrel{G_{j0} (j \neq i)}{\longrightarrow}x_ie_j \stackrel{{x_j}^{-1}G_{0j}}{\longrightarrow}e_0 = 1.

\delta=D+\widetilde{A}_{1}\left(a_{1}\right)+\widetilde{A}_{2}\left(a_{2}\right)+\widetilde{A}_{3}\left(a_{3}\right), \quad D \in \mathfrak{d}_{4}^{C}, a_{i} \in \mathfrak{C}^{C},

\delta = D + \widetilde{A}_1(a_1) + \widetilde{A}_2(a_2) + \widetilde{A}_3(a_3), \quad D \in {\text{\es {d}}_4}^C, a_i \in \text{\es {C}}^C,

\mathfrak{f}_{4}{ }^{C}=\mathfrak{d}_{4}{ }^{C} \oplus \widetilde{\mathfrak{A}}_{1}{ }^{C} \oplus \widetilde{\mathfrak{A}}_{2}{ }^{C} \oplus \widetilde{\mathfrak{A}}_{3}{ }^{C}=\mathfrak{d}_{4}{ }^{C} \oplus \widetilde{\mathfrak{A}}^{C} .

{\text{\es {f}}_4}^C = {\text{\es {d}}_4}^C \oplus \widetilde{\text{\es {A}}}_1^{\ C} \oplus \widetilde{\text{\es {A}}}_2^{\ C} \oplus \widetilde{\text{\es {A}}}_3^{\ C} = {\text{\es {d}}_4}^C \oplus \widetilde{\text{\es {A}}}^C.

\mathfrak{a} \ni\left[D^{\prime}, D+\sum_{i=1}^{3} \widetilde{A}_{i}\left(a_{i}\right)\right]=\left[D^{\prime}, D\right]+\sum_{i=1}^{3} \widetilde{A}_{i}\left(D_{i}{ }^{\prime} a_{i}\right)(\text { Lemma 2.4.2 }),

\text{\es {a}} \ni \Big[D', D + {\sum_{i=1}^3}\widetilde{A}_i(a_i)\Big] = [D',D] + {\sum_{i=1}^3}\widetilde{A}_i({D_i}'a_i)\; \text{(Lemma 2.4.2)},

\mathfrak{a} \supset\left[\mathfrak{a}, \mathfrak{f}_{4}{ }^{C}\right] \supset\left[\mathfrak{d}_{4}{ }^{C}, \widetilde{\mathfrak{A}}_{i}{ }^{C}\right]=\widetilde{\mathfrak{A}}_{i}{ }^{C}, \quad i=1,2,3 .

\text{\es {a}} \supset [\text{\es {a}}, {\text{\es {f}}_4}^C] \supset [{\text{\es {d}}_4}^C, \widetilde{\text{\es {A}}}_i^{\ C}] = \widetilde{\text{\es {A}}}_i^{\ C}, \quad i = 1, 2, 3.

\widetilde{A}_{1}\left(a_{1}\right)+\widetilde{A}_{2}\left(a_{2}\right)+\widetilde{A_{3}}\left(a_{3}\right) \in \widetilde{\mathfrak{A}}^{C} \cap \mathfrak{a} \subset \mathfrak{a} .

\widetilde{A}_1(a_1) + \widetilde{A}_2(a_2) + \widetilde{A_3}(a_3) \in \widetilde{\text{\es {A}}}^C \cap \text{\es {a}} \subset \text{\es {a}}.

\widetilde{A}_{1}(1)+\widetilde{A}_{2}(b)+\widetilde{A}_{3}(c) \in \mathfrak{a} .

\widetilde{A}_1(1) + \widetilde{A}_2(b) + \widetilde{A}_3(c) \in \text{\es {a}}.

0 \neq\left[\widetilde{A}_{1}(1), \widetilde{A}_{1}\left(e_{1}\right)\right] \in \mathfrak{d}_{4}{ }^{C} \cap \mathfrak{a}

0 \neq [\widetilde{A}_1(1), \widetilde{A}_1(e_1)] \in {\text{\es {d}}_4}^C \cap \text{\es {a}}

-\widetilde{A}_{3}(\bar{b})+\widetilde{A}_{2}(\bar{c}) \in \mathfrak{a} \quad \text { and then } \quad \widetilde{A}_{3}(1)+\widetilde{A}_{2}\left(c^{\prime}\right) \in \mathfrak{a}

-\widetilde{A}_3(\overline{b}) + \widetilde{A}_2(\overline{c}) \in \text{\es {a}} \quad \text{and then $\quad \widetilde{A}_3(1) + \widetilde{A}_2(c') \in \text{\es {a}}$}

[\delta,[\widetilde{A}, \widetilde{B}]]=[\widetilde{\delta A}, \widetilde{B}]+[\widetilde{A}, \widetilde{\delta B}] .

[\delta, [\widetilde{A}, \widetilde{B}]\,] = [\widetilde{\delta A}, \widetilde{B}] + [\widetilde{A}, \widetilde{\delta B}].

\begin{aligned} & =\delta(A \circ(B \circ X)-B \circ(A \circ X))-(A \circ(B \circ \delta X)-B \circ(A \circ \delta X)) \\ & =\delta A \circ(B \circ X)+A \circ(\delta B \circ X)+A \circ(B \circ \delta X)-\delta B \circ(A \circ X) \\ & -B \circ(\delta A \circ X)-B \circ(A \circ \delta X)-A \circ(B \circ \delta X)+B \circ(A \circ \delta X) \\ & =\delta A \circ(B \circ X)-B \circ(\delta A \circ X)+A \circ(\delta B \circ X)-\delta B \circ(A \circ X) \\ & =[\widetilde{\delta A}, \widetilde{B}] X+[\widetilde{A}, \widetilde{\delta B}] X, \quad X \in \mathfrak{J}^{C} . \end{aligned}

F_{i}(1) \xrightarrow{\widetilde{A}_{i}(1)} E_{i+1}-E_{i+2} \xrightarrow{\widetilde{A}_{i+2}(1)} F_{i+2}(1) \longrightarrow \cdots \longrightarrow F_{i+1}(1),

F_i(1) \;\stackrel{\widetilde{A}_i(1)}{\longrightarrow}\; E_{i+1} - E_{i+2} \stackrel{\widetilde{A}_{i+2}(1) }{\longrightarrow}\; F_{i+2}(1) {\longrightarrow}\; \cdots \;{\longrightarrow}\; F_{i+1}(1),

0 \neq X=\xi\left(E_{1}-E_{2}\right)+\eta\left(E_{2}-E_{3}\right)+F_{1}\left(x_{1}\right)+F_{2}\left(x_{2}\right)+F_{3}\left(x_{3}\right) \in W .

0 \neq X = \xi(E_1 - E_2) + \eta(E_2 - E_3) + F_1(x_1) + F_2(x_2) + F_3(x_3) \in W.

F_{1}(1)+F_{2}\left(y_{2}\right)+F_{3}\left(y_{3}\right) \in W .

F_1(1) + F_2(y_2) + F_3(y_3) \in W.

F_{3}(1)+F_{1}\left(z_{1}\right) \in W .

F_3(1) + F_1(z_1) \in W.

X=\xi\left(E_{1}-E_{2}\right)+\eta\left(E_{2}-E_{3}\right) \in W,

X = \xi(E_1 - E_2) + \eta(E_2 - E_3) \in W,

([\widetilde{A}, \widetilde{B}] C, D)=([\widetilde{C}, \widetilde{D}] A, B) .

([\widetilde{A}, \widetilde{B}]C, D) = ([\widetilde{C}, \widetilde{D}]A, B).

\begin{aligned} & ([\widetilde{A}, \widetilde{B}] C, D)=(A \circ(B \circ C), D)-(B \circ(A \circ C), D) \\ & \quad=(B \circ C, A \circ D)-(A \circ C, B \circ D) \\ & \quad=(C \circ(D \circ A), B)-(D \circ(C \circ A), B)=([\widetilde{C}, \widetilde{D}] A, B) . \end{aligned}

(\delta,[\widetilde{A}, \widetilde{B}])_{4}=(\delta A, B), \quad \delta \in \mathfrak{f}_{4}{ }^{C}, \quad A, B \in \mathfrak{J}^{C} .

(\delta, [\widetilde{A}, \widetilde{B}])_4 = (\delta A, B), \quad \delta \in {\text{\es {f}}_4}^C,\;\;A, B \in \text{\es {J}}^C.

\left(\delta_{1}, \delta_{2}\right)_{4}=\sum_{i, j}\left(\left[\widetilde{A}_{i}, \widetilde{B}_{i}\right] C_{j}, D_{j}\right)=\sum_{i, j}\left(\left[\widetilde{C}_{j}, \widetilde{D}_{j}\right] A_{i}, B_{i}\right),

(\delta_1, \delta_2)_4 = {\sum_{i,j}}([\widetilde{A}_i, \widetilde{B}_i]C_j, D_j) = {\sum_{i,j}}([\widetilde{C}_j, \widetilde{D}_j]A_i, B_i),

\left(\left[\delta, \delta_{1}\right], \delta_{2}\right)_{4}+\left(\delta_{1},\left[\delta, \delta_{2}\right]\right)_{4}=0, \quad \delta, \delta_{i} \in \mathfrak{f}_{4}^{C} .

([\delta, \delta_1], \delta_2)_4 + (\delta_1, [\delta, \delta_2])_4 = 0, \quad \delta, \delta_i \in {\text{\es {f}}_4}^C.

\begin{aligned} & ([\delta,[\widetilde{A}, \widetilde{B}]],[\widetilde{C}, \widetilde{D}])_{4}+([\widetilde{A}, \widetilde{B}],[\delta,[\widetilde{C}, \widetilde{D}]])_{4} \\ & =([\widetilde{\delta A}, \widetilde{B}]+[\widetilde{A}, \widetilde{\delta B}],[\widetilde{C}, \widetilde{D}])_{4}+([\widetilde{A}, \widetilde{B}],[\widetilde{\delta C}, \widetilde{D}]+[\widetilde{C}, \widetilde{\delta D}])_{4}(\text { Lemma 2.4.5.(1) }) \\ = & ([\widetilde{\delta A}, \widetilde{B}] C, D)+([\widetilde{A}, \widetilde{\delta B}] C, D)+([\widetilde{A}, \widetilde{B}] \delta C, D)+([\widetilde{A}, \widetilde{B}] C, \delta D) \\ = & (\delta A \circ(B \circ C), D)-(B \circ(\delta A \circ C), D)+(A \circ(\delta B \circ C), D)-(\delta B \circ(A \circ C), D) \\ & +(A \circ(B \circ \delta C), D)-(B \circ(A \circ \delta C), D)+(A \circ(B \circ C), \delta D)-(B \circ(A \circ C), \delta D) \end{aligned}

\begin{array}{l} ([\delta, [\widetilde{A}, \widetilde{B}]\,], [\widetilde{C}, \widetilde{D}])_4 + ([\widetilde{A}, \widetilde{B}], [\delta, [\widetilde{C}, \widetilde{D}]\,])_4 \vspace{1mm}\\ \;\;= ([\widetilde{\delta A}, \widetilde{B}] + [\widetilde{A}, \widetilde{\delta B}], [\widetilde{C}, \widetilde{D}])_4 + ([\widetilde{A}, \widetilde{B}], [\widetilde{\delta C}, \widetilde{D}] + [\widetilde{C}, \widetilde{\delta D}])_4 \;\; \text{(Lemma 2.4.5.(1))} \vspace{1mm}\\ \;\;= ([\widetilde{\delta A}, \widetilde{B}]C, D) + ([\widetilde{A}, \widetilde{\delta B}]C, D) + ([\widetilde{A}, \widetilde{B}]\delta C, D) + ([\widetilde{A}, \widetilde{B}]C, \delta D) \vspace{1mm}\\ \;\;= (\delta A \circ (B \circ C), D) - (B \circ (\delta A \circ C), D) + (A \circ (\delta B \circ C), D) - (\delta B \circ (A \circ C), D) \vspace{1mm}\\ \;\;\;\;\;+ (A \circ (B \circ \delta C), D) - (B \circ (A \circ \delta C), D) + (A \circ (B \circ C), \delta D) - (B \circ (A \circ C), \delta D), \end{array}

B_{4}\left(\delta_{1}, \delta_{2}\right)=9\left(\delta_{1}, \delta_{2}\right)_{4}=3 \operatorname{tr}\left(\delta_{1} \delta_{2}\right), \quad \delta_{1}, \delta_{2} \in \mathfrak{f}_{4}^{C} .

B_4(\delta_1, \delta_2) = 9(\delta_1, \delta_2)_4 = 3\text\mathrm{{tr}}(\delta_1\delta_2), \quad \delta_1, \delta_2 \in {\text{\es {f}}_4}^C.

B_{4}\left(\delta_{1}, \delta_{2}\right)=k\left(\delta_{1}, \delta_{2}\right)_{4}=k^{\prime} \operatorname{tr}\left(\delta_{1} \delta_{2}\right) .

B_4(\delta_1, \delta_2) = k(\delta_1, \delta_2)_4 = k'\text\mathrm{{tr}}(\delta_1\delta_2).

\begin{aligned} (\delta, \delta)_{4} & =\left(\widetilde{A}_{1}(1), \widetilde{A}_{1}(1)\right)_{4}=-2\left(\widetilde{A}_{1}(1),\left[\widetilde{E}_{3}, \widetilde{F}_{1}(1)\right]\right)_{4} \\ & =-2\left(\widetilde{A}_{1}(1) E_{3}, F_{1}(1)\right)=-\left(F_{1}(1), F_{1}(1)\right)=-2 . \end{aligned}

\begin{aligned} (\delta, \delta)_4 \!\!\! &=& \!\!\! (\widetilde{A}_1(1), \widetilde{A}_1(1))_4 = -2(\widetilde{A}_1(1), [\widetilde{E}_3, \widetilde{F}_1(1)])_4 \\ \!\!\! &=& \!\!\! -2(\widetilde{A}_1(1)E_3, F_1(1)) = -(F_1(1), F_1(1)) = - 2. \end{aligned}

\begin{aligned} & {\left[\widetilde{A}_{1}(1),\left[\widetilde{A}_{1}(1), G_{i 0}\right]\right]=-\left[\widetilde{A}_{1}(1), \widetilde{A}_{1}\left(e_{i}\right)\right]=-G_{i 0}, \quad i \neq 0} \\ & {\left[\widetilde{A}_{1}(1),\left[\widetilde{A}_{1}(1), \widetilde{A}_{1}\left(e_{i}\right)\right]\right]=\left[\widetilde{A}_{1}(1), G_{i 0}\right]=-\widetilde{A}_{1}\left(e_{i}\right), \quad i \neq 0} \\ & {\left[\widetilde{A}_{1}(1),\left[\widetilde{A}_{1}(1), \widetilde{A}_{2}\left(e_{i}\right)\right]\right]=\frac{1}{2}\left[\widetilde{A}_{1}(1), \widetilde{A}_{3}\left(e_{i}\right)\right]=-\frac{1}{4} \widetilde{A}_{2}\left(e_{i}\right),} \\ & {\left[\widetilde{A}_{1}(1),\left[\widetilde{A}_{1}(1), \widetilde{A}_{3}\left(e_{i}\right)\right]\right]=-\frac{1}{2}\left[\widetilde{A}_{1}(1), \widetilde{A}_{2}\left(e_{i}\right)\right]=-\frac{1}{4} \widetilde{A}_{3}\left(e_{i}\right),} \\ & \text { the others }=0 . \end{aligned}

\begin{array}{l} [\widetilde{A}_1(1), [\widetilde{A}_1(1), G_{i0}]\,] = -[\widetilde{A}_1(1), \widetilde{A}_1(e_i)] = -G_{i0}, \;\; i \neq 0 \vspace{1mm}\\ {[}\widetilde{A}_1(1), [\widetilde{A}_1(1), \widetilde{A}_1(e_i)]\,{]} = [\widetilde{A}_1(1), G_{i0}] = - \widetilde{A}_1(e_i),\;\; i \neq 0 \vspace{1mm}\\ {[}\widetilde{A}_1(1), [\widetilde{A}_1(1), \widetilde{A}_2(e_i)]\,{]} = {\frac{1}{2}}[\widetilde{A}_1(1), \widetilde{A}_3(e_i)] = - {\frac{1}{4}}\widetilde{A}_2(e_i), \vspace{1mm}\\ {[}\widetilde{A}_1(1), [\widetilde{A}_1(1), \widetilde{A}_3(e_i)]\,{]} = - {\frac{1}{2}}[\widetilde{A}_1(1), \widetilde{A}_2(e_i)] = - {\frac{1}{4}}\widetilde{A}_3(e_i), \vspace{1mm}\\ \text{the others} \; = 0. \end{array}

B_{4}(\delta, \delta)=\operatorname{tr}\left(\left(\operatorname{ad} \widetilde{A}_{1}(1)\right)^{2}\right)=(-1) \times 7 \times 2+\left(-\frac{1}{4}\right) \times 8 \times 2=-18 .

B_4(\delta, \delta) = \text\mathrm{{tr}}((\text{ad}\widetilde{A}_1(1))^2) = (-1) \times 7 \times 2 + \Big(-\frac{1}{4} \Big) \times 8 \times 2 = -18.

\begin{aligned} & \widetilde{A}_{1}(1) \widetilde{A}_{1}(1) E_{2}=-\frac{1}{2} \widetilde{A}_{1}(1) F_{1}(1)=-\frac{1}{2}\left(E_{2}-E_{3}\right), \\ & \widetilde{A}_{1}(1) \widetilde{A}_{1}(1) E_{3}=\frac{1}{2} \widetilde{A}_{1}(1) F_{1}(1)=\frac{1}{2}\left(E_{2}-E_{3}\right), \\ & \widetilde{A}_{1}(1) \widetilde{A}_{1}(1) F_{1}(1)=\widetilde{A}_{1}(1)\left(E_{2}-E_{3}\right)=-F_{1}(1), \\ & \widetilde{A}_{1}(1) \widetilde{A}_{1}(1) F_{2}\left(e_{i}\right)=\frac{1}{2} \widetilde{A}_{1}(1) F_{3}\left(e_{i}\right)=-\frac{1}{4} F_{2}\left(e_{i}\right), \\ & \widetilde{A}_{1}(1) \widetilde{A}_{1}(1) F_{3}\left(e_{i}\right)=-\frac{1}{2} \widetilde{A}_{1}(1) F_{3}\left(\overline{e_{i}}\right)=-\frac{1}{4} F_{3}\left(e_{i}\right), \\ & \text { the others }=0 . \end{aligned}

\begin{array}{l} \widetilde{A}_1(1)\widetilde{A}_1(1)E_2 = - {\frac{1}{2}}\widetilde{A}_1(1)F_1(1) = - {\frac{1}{2}}(E_2 - E_3), \vspace{1mm} \\ \widetilde{A}_1(1)\widetilde{A}_1(1)E_3 = {\frac{1}{2}}\widetilde{A}_1(1)F_1(1) = {\frac{1}{2}}(E_2 - E_3), \vspace{1mm}\\ \widetilde{A}_1(1)\widetilde{A}_1(1)F_1(1) = \widetilde{A}_1(1)(E_2 - E_3) = - F_1(1), \vspace{1mm}\\ \widetilde{A}_1(1)\widetilde{A}_1(1)F_2(e_i) = {\frac{1}{2}}\widetilde{A}_1(1)F _3(\overline{e_i}) = - {\frac{1}{4}}F_2(e_i), \vspace{1mm}\\ \widetilde{A}_1(1)\widetilde{A}_1(1)F_3(e_i) = - {\frac{1}{2}}\widetilde{A}_1(1)F_3(\overline{e_i}) = - {\frac{1}{4}}F_3(e_i), \vspace{1mm}\\ \text{the others}\; = 0. \end{array}

\operatorname{tr}(\delta \delta)=\operatorname{tr}\left(\left(\widetilde{A}_{1}(1)\right)^{2}\right)=\left(-\frac{1}{2}\right) \times 2-1+\left(-\frac{1}{4}\right) \times 8 \times 2=-6 .

\text\mathrm{{tr}}(\delta\delta) = \text\mathrm{{tr}}((\widetilde{A}_1(1))^2) = \Big(-{\frac{1}{2}}\Big) \times 2 - 1 + \Big(-{\frac{1}{4}}\Big) \times 8 \times 2 = - 6.

\mathfrak{D}_{4}{ }^{C}=\left\{D \in \operatorname{Hom}_{C}\left(\mathfrak{C}^{C}\right) \mid(D x, y)+(x, D y)=0\right\} .

{\text{\es {D}}_4}^C = \{ D \in \text\mathrm{{Hom}}_C(\text{\es {C}}^C) \, | \, (Dx, y) + (x, Dy) = 0 \}.

\mathfrak{h}=\left\{H=\sum_{k=0}^{3} \lambda_{i} H_{k} \mid \lambda_{k} \in C\right\}

\text{\es {h}} = \{ H = \sum_{k=0}^3\lambda_iH_k \, | \, \lambda_k \in C \}

\pm\left(\lambda_{k}-\lambda_{l}\right), \quad \pm\left(\lambda_{k}+\lambda_{l}\right), \quad 0 \leq k<l \leq 3 .

\pm (\lambda_k - \lambda_l),\;\; \pm (\lambda_k + \lambda_l), \quad 0 \le k < l \le 3.

\begin{aligned} \lambda_{k}-\lambda_{l} & :\left(G_{k l}+G_{4+k 4+l}\right)-i\left(G_{k 4+l}+G_{l 4+k}\right), \\ -\lambda_{k}+\lambda_{l} & : i\left(G_{k l}+G_{4+k 4+l}\right)-\left(G_{k 4+l}+G_{l 4+k}\right), \\ \lambda_{k}+\lambda_{l} & :\left(G_{k l}-G_{4+k 4+l}\right)+i\left(G_{k 4+l}-G_{l 4+k}\right), \\ -\lambda_{k}-\lambda_{l} & :\left(G_{k l}-G_{4+k 4+l}\right)+\left(G_{k 4+l}-G_{l 4+k}\right) . \end{aligned}

\begin{aligned} \;\;\;\lambda_k - \lambda_l &:& (G_{kl} + G_{4+k 4+l}) - i(G_{k 4+l} + G_{l 4+k}), \vspace{1mm}\\ -\lambda_k + \lambda_l &:& i(G_{k l} + G_{4+k 4+l}) - (G_{k 4+l} + G_{l 4+k}), \vspace{1mm}\\ \;\;\;\lambda_k + \lambda_l &:& (G_{kl} - G_{4+k 4+l}) + i(G_{k 4+l} - G_{l 4+k}), \vspace{1mm}\\ -\lambda_k - \lambda_l &:& i(G_{k l} - G_{4+k 4+l}) + (G_{k 4+l} - G_{l 4+k}). \end{aligned}

\mathfrak{D}_{4}{ }^{C} \ni D \rightarrow(D, \nu D, \kappa \pi D) \in \mathfrak{d}_{4}{ }^{C} \subset \mathfrak{f}_{4}{ }^{C}

{\text{\es {D}}_4}^C \ni D \; \to \; (D, \nu D, \kappa\pi D) \in {\text{\es {d}}_4}^C \subset {\text{\es {f}}_4}^C

\nu=\pi \kappa=\frac{1}{2}\left(\begin{array}{rrrr} -1 & -1 & 1 & -1 \\ 1 & 1 & 1 & -1 \\ -1 & 1 & 1 & 1 \\ 1 & -1 & 1 & 1 \end{array}\right), \quad \pi=\frac{1}{2}\left(\begin{array}{rrrr} 1 & -1 & 1 & -1 \\ -1 & 1 & 1 & -1 \\ 1 & 1 & 1 & 1 \\ -1 & -1 & 1 & 1 \end{array}\right)

\begin{aligned} \pm\left(\lambda_{k}-\lambda_{l}\right), & \pm\left(\lambda_{k}+\lambda_{l}\right), \quad 0 \leq k<l \leq 3, \\ \pm \lambda_{0}, \quad \pm \lambda_{1}, \quad & \pm \lambda_{2}, \quad \pm \lambda_{3}, \\ \pm \frac{1}{2}\left(-\lambda_{0}-\lambda_{1}+\lambda_{2}-\lambda_{3}\right), & \pm \frac{1}{2}\left(\lambda_{0}+\lambda_{1}+\lambda_{2}-\lambda_{3}\right), \\ \pm \frac{1}{2}\left(-\lambda_{0}+\lambda_{1}+\lambda_{2}+\lambda_{3}\right), & \pm \frac{1}{2}\left(\lambda_{0}-\lambda_{1}+\lambda_{2}+\lambda_{3}\right), \\ \pm \frac{1}{2}\left(\lambda_{0}-\lambda_{1}+\lambda_{2}-\lambda_{3}\right), & \pm \frac{1}{2}\left(-\lambda_{0}+\lambda_{1}+\lambda_{2}-\lambda_{3}\right), \\ \pm \frac{1}{2}\left(\lambda_{0}+\lambda_{1}+\lambda_{2}+\lambda_{3}\right), & \pm \frac{1}{2}\left(-\lambda_{0}-\lambda_{1}+\lambda_{2}+\lambda_{3}\right) . \end{aligned}

\mathfrak{f}_{4}{ }^{C}=\mathfrak{d}_{4}{ }^{C} \oplus \widetilde{\mathfrak{A}}_{1}{ }^{C} \oplus \widetilde{\mathfrak{A}}_{2}{ }^{C} \oplus \widetilde{\mathfrak{A}}_{3}{ }^{C} .

{\text{\es {f}}_4}^C = {\text{\es {d}}_4}^C \oplus \widetilde{\text{\es {A}}}_1^{\ C} \oplus \widetilde{\text{\es {A}}}_2^{\ C} \oplus \widetilde{\text{\es {A}}}_3^{\ C}.

\pm\left(\lambda_{k}-\lambda_{l}\right), \quad \pm\left(\lambda_{k}+\lambda_{l}\right), \quad 0 \leq k<l \leq 3

\begin{array}{c} \pm (\lambda_k - \lambda_l), \quad \pm (\lambda_k + \lambda_l) \;\;,\;\; 0 \le k < l \le 3, \vspace{1mm}\\ \pm \lambda_0, \quad \pm \lambda_1, \quad \pm \lambda_2, \quad \pm \lambda_3, \end{array}

\begin{aligned} H\left(e_{k}+i e_{4+k}\right) & =-\sum_{j=0}^{3} \lambda_{k} i G_{j 4+j}\left(e_{k}+i e_{4+k}\right) \\ & =-i \lambda_{k}\left(-e_{4+k}+i e_{k}\right)=\lambda_{k}\left(e_{k}+i e_{4+k}\right), \end{aligned}

\begin{aligned} H(e_k + ie_{4+k}) \!\!\! &=& \!\!\! - \sum_{j=0}^3\lambda_kiG_{j4+j}(e_k + ie_{4+k}) \\ \!\!\! &=& \!\!\! -i\lambda_k(- e_{4+k} + ie_k) = \lambda_k(e_{k} + ie_{4+k}), \end{aligned}

\begin{aligned} \nu H= & \nu\left(\sum_{k=0}^{3} \lambda_{k} H_{k}\right) \\ = & \frac{1}{2}\left(\lambda_{0}\left(-H_{0}+H_{1}-H_{2}+H_{3}\right)+\lambda_{1}\left(-H_{0}+H_{1}+H_{2}-H_{3}\right)\right. \\ & \left.+\lambda_{2}\left(H_{0}+H_{1}+H_{2}+H_{3}\right)+\lambda_{3}\left(-H_{0}-H_{1}+H_{2}+H_{3}\right)\right) \\ = & \frac{1}{2}\left(-\lambda_{0}-\lambda_{1}+\lambda_{2}-\lambda_{3}\right) H_{0}+\frac{1}{2}\left(\lambda_{0}+\lambda_{1}+\lambda_{2}-\lambda_{3}\right) H_{1} \\ & +\frac{1}{2}\left(-\lambda_{0}+\lambda_{1}+\lambda_{2}+\lambda_{3}\right) H_{2}+\frac{1}{2}\left(\lambda_{0}-\lambda_{1}+\lambda_{2}+\lambda_{3}\right) H_{3}, \end{aligned}

\begin{array}{l} \nu H = \nu\Big(\sum_{k=0}^3\lambda_kH_k\Big) \vspace{1mm}\\ \quad \;\;\; = \frac{1}{2}(\lambda_0(- H_0 + H_1 - H_2 + H_3) + \lambda_1(- H_0 + H_1 + H_2 - H_3) \vspace{1mm}\\ \qquad \;\; + \lambda_2(H_0 + H_1 + H_2 + H_3) + \lambda_3(- H_0 - H_1 + H_2 +H _3)) \vspace{1mm}\\ \quad \;\;\; = \frac{1}{2}(- \lambda_0 - \lambda_1 + \lambda_2 - \lambda_3)H_0 + \frac{1}{2}(\lambda_0 + \lambda_1 + \lambda_2 - \lambda_3)H_1 \vspace{1mm}\\ \qquad \;\; + \frac{1}{2}(- \lambda_0 + \lambda_1 + \lambda_2 + \lambda_3)H_2 + \frac{1}{2}(\lambda_0 - \lambda_1 + \lambda_2 + \lambda_3)H_3, \end{array}

\begin{aligned} \kappa \pi H= & \frac{1}{2}\left(-\lambda_{0}+\lambda_{1}-\lambda_{2}+\lambda_{3}\right) H_{0}+\frac{1}{2}\left(-\lambda_{0}+\lambda_{1}+\lambda_{2}-\lambda_{3}\right) H_{1} \\ & +\frac{1}{2}\left(\lambda_{0}+\lambda_{1}+\lambda_{2}+\lambda_{3}\right) H_{2}+\frac{1}{2}\left(-\lambda_{0}-\lambda_{1}+\lambda_{2}+\lambda_{3}\right) H_{3} \end{aligned}

\alpha_{1}=\lambda_{0}-\lambda_{1}, \quad \alpha_{2}=\lambda_{1}-\lambda_{2}, \quad \alpha_{3}=\lambda_{2}, \quad \alpha_{4}=\frac{1}{2}\left(-\lambda_{0}-\lambda_{1}-\lambda_{2}+\lambda_{3}\right)

\alpha_1 = \lambda_0 - \lambda_1, \;\; \alpha_2 = \lambda_1 - \lambda_2,\;\; \alpha_3 = \lambda_2,\;\; \alpha_4 = \frac{1}{2}(- \lambda_0 - \lambda_1 - \lambda_2 + \lambda_3)

\mu=2 \alpha_{1}+3 \alpha_{2}+4 \alpha_{3}+2 \alpha_{4}

\mu = 2\alpha_1 + 3\alpha_2 + 4\alpha_3 + 2\alpha_4

\begin{array}{lr} \lambda_{0}=\alpha_{1}+\alpha_{2}+ & \alpha_{3} \\ \lambda_{1}= & \alpha_{2}+\alpha_{3} \\ \lambda_{3}= & \alpha_{3} \\ \lambda_{3}= & \alpha_{1}+2 \alpha_{2}+3 \alpha_{3}+2 \alpha_{4}, \end{array}

\begin{array}{llrrrrrrr} \lambda_0 \!\!\! &= \alpha_1 \!\!\!\! &+& \!\!\!\! \alpha_2 \!\!\!\! &+& \!\!\!\! \alpha_3 && \vspace{1mm}\\ \lambda_1 \!\!\! &= && \!\!\!\! \alpha_2 \!\!\!\! &+& \!\!\!\!\alpha_3 && \vspace{1mm}\\ \lambda_3 \!\!\! &= && \!\!\!\! & \!\!\!\! & \!\!\!\! \alpha_3 && \vspace{1mm}\\ \lambda_3 \!\!\! &= \alpha_1 \!\!\!\! &+& \!\!\!\! 2\alpha_2 \!\!\!\! &+& \!\!\!\! 3\alpha_3& \!\!\!\! +& \!\!\!\! 2\alpha_4, \end{array}

\mathfrak{h}_{\boldsymbol{R}}=\left\{H=\sum_{k=0}^{3} \lambda_{k} H_{k} \mid \lambda_{k} \in \boldsymbol{R}\right\} .

\text{\es {h}}_{\text{${R}$}} = \{H = \sum_{k=0}^3\lambda_kH_k \, | \, \lambda_k \in \text{$R$} \}.

B_{4}\left(H, H^{\prime}\right)=18 \sum_{k=0}^{3} \lambda_{k} \lambda_{k}{ }^{\prime}, \quad H=\sum_{k=0}^{3} \lambda_{k} H_{k}, H^{\prime}=\sum_{k=0}^{3} \lambda_{k}{ }^{\prime} H_{k} \in \mathfrak{h}_{\boldsymbol{R}} .

B_4(H,H') = 18\sum_{k=0}^3\lambda_k{\lambda_k}', \quad H = \sum_{k=0}^3\lambda_kH_k, H' = \sum_{k=0}^3{\lambda_k}'H_k \in \text{\es {h}}_{\text{${R}$}}.

\begin{aligned} & H E_{i}=0, \quad i=1,2,3 \\ & H F_{1}(x)=F_{1}(H x), H F_{2}(x)=F_{2}((\nu H) x), H F_{3}(x)=F_{3}((\kappa \pi H) x), \end{aligned}

\begin{array}{ll} HE_i = 0, \quad i = 1, 2, 3, \vspace{1mm}\\ HF_1(x) = F_1(Hx), \, \, HF_2(x) = F_2((\nu H)x), \, \, HF_3(x) = F_3((\kappa\pi H)x), \end{array}

\begin{aligned} & B_{4}\left(H, H^{\prime}\right)=\sum_{k, l=0}^{3} \lambda_{k} \lambda_{l}^{\prime} B_{4}\left(H_{k}, H_{l}\right) \\ & \quad=3 \sum_{k, l}\left(\lambda_{k} \lambda_{l}^{\prime}\left(\operatorname{tr}\left(H_{k} H_{l}\right)+\operatorname{tr}\left(\left(\nu H_{k}\right)\left(\nu H_{l}\right)\right)+\operatorname{tr}\left(\left(\kappa \pi H_{k}\right)\left(\kappa \pi H_{l}\right)\right)\right)\right) \\ & \quad=3 \sum_{k, l} \lambda_{k} \lambda_{l}^{\prime}(2+2+2) \delta_{k l}=18 \sum_{k=0}^{3} \lambda_{k} \lambda_{k}^{\prime} \end{aligned}

\begin{array}{ll} H_{\alpha_{1}}=\frac{1}{18}\left(H_{0}-H_{1}\right), & H_{\alpha_{2}}=\frac{1}{18}\left(H_{1}-H_{2}\right), \\ H_{\alpha_{3}}=\frac{1}{18} H_{2}, & H_{\alpha_{4}}=\frac{1}{36}\left(-H_{0}-H_{1}-H_{2}+H_{3}\right) . \end{array}

\begin{array}{ll} H_{\alpha_1} = {\frac{1}{18}}(H_0 - H_1), & H_{\alpha_2} = {\frac{1}{18}}(H_1 - H_2), \vspace{1mm}\\ H_{\alpha_3} = {\frac{1}{18}}H_2,& H_{\alpha_4} = {\frac{1}{36}}(- H_0 - H_1 - H_2 + H_3). \end{array}

\left(\alpha_{1}, \alpha_{1}\right)=B_{4}\left(H_{\alpha_{1}}, H_{\alpha_{1}}\right)=18 \frac{1}{18} \frac{1}{18}(1+1)=\frac{1}{9}

(\alpha_1, \alpha_1) = B_4(H_{\alpha_1}, H_{\alpha_1}) = 18\frac{1}{18}\frac{1}{18}(1 + 1) = \frac{1}{9}

\begin{aligned} & \left(\alpha_{1}, \alpha_{1}\right)=\left(\alpha_{2}, \alpha_{2}\right)=\frac{1}{9}, \quad\left(\alpha_{3}, \alpha_{3}\right)=\left(\alpha_{4}, \alpha_{4}\right)=\frac{1}{18} \\ & \left(\alpha_{1}, \alpha_{2}\right)=-\frac{1}{18}, \quad\left(\alpha_{2}, \alpha_{3}\right)=-\frac{1}{18}, \quad\left(\alpha_{3}, \alpha_{4}\right)=-\frac{1}{36} \\ & \left(\alpha_{1}, \alpha_{3}\right)=\left(\alpha_{1}, \alpha_{4}\right)=\left(\alpha_{2}, \alpha_{4}\right)=0, \\ & (-\mu,-\mu)=\frac{1}{9}, \quad\left(-\mu, \alpha_{1}\right)=-\frac{1}{18}, \quad\left(-\mu, \alpha_{i}\right)=0, \quad i=2,3,4 \end{aligned}

\varphi\left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right)\left(\begin{array}{ccc} \xi_{1} & x_{3} & \bar{x}_{2} \\ \bar{x}_{3} & \xi_{2} & x_{1} \\ x_{2} & \bar{x}_{1} & \xi_{3} \end{array}\right)=\left(\begin{array}{ccc} \xi_{1} & \alpha_{3} x_{3} & \overline{\alpha_{2} x_{2}} \\ \overline{\alpha_{3} x_{3}} & \xi_{2} & \alpha_{1} x_{1} \\ \alpha_{2} x_{2} & \overline{\alpha_{1} x_{1}} & \xi_{3} \end{array}\right) .

\begin{aligned} & R\left(\left(\alpha_{1} x_{1}\right)\left(\alpha_{2} x_{2}\right)\left(\alpha_{3} x_{3}\right)\right)=R\left(\left(\overline{\alpha_{3}\left(\overline{x_{1} x_{2}}\right)}\right) \alpha_{3} x_{3}\right) \\ & \quad=\left(\alpha_{3}\left(\overline{x_{1} x_{2}}\right), \alpha_{3} x_{3}\right)=\left(\overline{x_{1} x_{2}}, x_{3}\right)=R\left(x_{1} x_{2} x_{3}\right) \end{aligned}

\begin{array}{l} R((\alpha_1x_1)(\alpha_2x_2)(\alpha_3x_3)) = R((\overline{\alpha_3(\overline{x_1x_2})})\alpha_3x_3) \vspace{1mm}\\ \qquad \quad = (\alpha_3(\overline{x_1x_2}), \alpha_3x_3) = (\overline{x_1x_2}, x_3) = R(x_1x_2x_3),\end{array}

\mathfrak{J}_{i}=\left\{F_{i}(x) \mid x \in \mathfrak{C}\right\}=\left\{X \in \mathfrak{J} \mid 2 E_{i+1} \circ X=2 E_{i+2} \circ X=X\right\}, \quad i=1,2,3 .

\text{\es {J}}_i = \{ F_i(x) \, | \, x \in \text{\es {C}} \} = \{ X \in \text{\es {J}} \, | \, 2E_{i+1} \circ X = 2E_{i+2} \circ X = X \}, \quad i = 1, 2, 3.

\alpha F_{i}(x)=F_{i}\left(\alpha_{i} x\right), \quad x \in \mathfrak{C},

\alpha F_i(x) = F_i(\alpha_i x), \quad x \in \text{\es {C}},

\left(\alpha_{i} x, \alpha_{i} y\right)=(x, y), \quad x, y \in \mathfrak{C} .

(\alpha_ix, \alpha_iy) = (x, y), \quad x,y \in \text{\es {C}}.

\left(\alpha_{1} x\right)\left(\alpha_{2} y\right)=\overline{\alpha_{3}(\overline{x y})}, \quad x, y \in \mathfrak{C} .

(\alpha_1x)(\alpha_2y) = \overline{\alpha_3(\overline{xy})}, \quad x,y \in \text{\es {C}}.

\left(F_{4}\right)_{E_{1}}=\left\{\alpha \in F_{4} \mid \alpha E_{1}=E_{1}\right\} .

(F_4)_{E_1} = \{ \alpha \in F_4 \, | \, \alpha E_1 = E_1 \}.

\mathfrak{J}_{01}=\left\{X \in \mathfrak{J} \mid E_{1} \circ X=0, \operatorname{tr}(X)=0\right\}=\left\{\left.\left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & \xi & x \\ 0 & \bar{x} & -\xi \end{array}\right) \right\rvert\, \xi \in \boldsymbol{R}, x \in \mathfrak{C}\right\},

\text{\es {J}}_{01} = \{ X \in \text{\es {J}} \, | \, E_1 \circ X = 0, \text\mathrm{{tr}}(X) = 0 \} = \Bigl\{\pmatrix{ 0 & 0 & 0 \cr 0 & \xi & x \cr 0 & \overline{x} & - \xi} \, \Big| \, \xi \in \text{$R$}, x \in \text{\es {C}} \Bigl\},

\mathfrak{J}_{23}=\left\{Y \in \mathfrak{J} \mid 2 E_{1} \circ Y=Y\right\}=\left\{\left.\left(\begin{array}{ccc} 0 & y_{3} & \bar{y}_{2} \\ \bar{y}_{3} & 0 & 0 \\ y_{2} & 0 & 0 \end{array}\right) \right\rvert\, y_{2}, y_{3} \in \mathfrak{C}\right\} .

\text{\es {J}}_{23} = \{ Y \in \text{\es {J}} \, | \, 2E_1 \circ Y = Y \} = \Bigl\{\pmatrix{0 & y_3 & \overline{y}_2 \cr \overline{y}_3 & 0 & 0 \cr y_2 & 0 & 0} \, \Big| \, y_2,y_3 \in \text{\es {C}} \Bigl\}.

Z_{0}=2 X_{0} \circ Y_{0} .

Z_0 = 2X_0 \circ Y_0.

Y_{1}=-2 Z_{0} \circ X_{1}, \quad Z_{2}=-2 X_{2} \circ Y_{0}, \quad X_{3}=-2 Y_{1} \circ Z_{2} .

Y_1 = - 2Z_0 \circ X_1, \quad Z_2 = -2X_2 \circ Y_0, \quad X_3 = - 2Y_1 \circ Z_2.

\begin{array}{lll} Z_{4}=-2 X_{4} \circ Y_{0}, & Y_{2}=-2 Z_{0} \circ X_{2}, & Y_{3}=-2 Z_{0} \circ X_{3}, \\ X_{5}=-2 Y_{1} \circ Z_{4}, & X_{6}=2 Y_{2} \circ Z_{4}, & X_{7}=-2 Y_{3} \circ Z_{4} \end{array}

\begin{array}{lll} Z_4 = - 2X_4 \circ Y_0, & Y_2 = - 2Z_0 \circ X_2, & Y_3 = - 2Z_0 \circ X_3, \vspace{1mm}\\ X_5 = -2Y_1 \circ Z_4, & X_6 = 2Y_2 \circ Z_4, & X_7 = - 2Y_3 \circ Z_4 \end{array}

\begin{array}{ll} Y_{i}=-2 Z_{0} \circ X_{i}, & i=4,5,6,7, \\ Z_{i}=-2 X_{i} \circ Y_{0}, & i=1,3,5,6,7 . \end{array}

\begin{array}{l} Y_i = - 2Z_0 \circ X_i, \quad i = 4, 5, 6, 7, \vspace{1mm}\\ Z_i = -2X_i \circ Y_0, \quad i= 1, 3, 5, 6, 7. \end{array}

\begin{array}{lll} \alpha E=E, & \alpha E_{1}=E_{1}, & \alpha\left(E_{2}-E_{3}\right)=A, \\ \alpha F_{1}\left(e_{i}\right)=X_{i}, & \alpha F_{2}\left(e_{i}\right)=Y_{i}, & \alpha F_{3}\left(e_{i}\right)=Z_{i}, \quad i=0,1, \cdots, 7 \end{array}

\begin{array}{lll} \alpha E = E, & \alpha E_1 = E_1, & \alpha(E_2 - E_3) = A, \vspace{1mm}\\ \alpha F_1(e_i) = X_i, & \alpha F_2(e_i) = Y_i, & \alpha F_3(e_i) = Z_i, \quad i = 0, 1,\cdots,7 \end{array}

\alpha(X \circ Y)=\alpha X \circ \alpha Y, \quad X, Y \in \mathfrak{J} .

\alpha(X \circ Y) = \alpha X \circ \alpha Y, \quad X, Y \in \text{\es {J}}.

p:\left(F_{4}\right)_{E_{1}} \rightarrow S O(9) .

p : (F_4)_{E_1} \to SO(9).

\begin{array}{clcccccc} 1 & \longrightarrow & \operatorname{Spin}(8) & \longrightarrow & \left(F_{4}\right)_{E_{1}} & \longrightarrow & S^{8} & \longrightarrow \\ & & \downarrow p^{\prime} & & \downarrow p & & \downarrow= & \\ 1 & \longrightarrow & S O(8) & \longrightarrow & S O(9) & \longrightarrow & S^{8} & \longrightarrow \end{array} .

\left(F_{4}\right)_{E_{1}} /\{1, \sigma\} \cong S O(9) .

(F_4)_{E_1}/\{1,\sigma\} \cong SO(9).

f\left(\alpha^{\prime}\right)=\varepsilon \alpha^{\prime} \varepsilon^{-1}, \quad(\epsilon x=\bar{x}, x \in \mathfrak{C})

f(\alpha') = \varepsilon\alpha'\varepsilon^{-1}, \quad \text{($\epsilon x = \overline{x}, x \in \text{\es {C}}$)}

\begin{aligned} & \left\{\begin{array}{l} \eta_{1}=\xi_{1} \\ \eta_{2}=\frac{\xi_{2}+\xi_{3}}{2}+\frac{\xi_{2}-\xi_{3}}{2} \cos 2|a|+\frac{\left(a, x_{1}\right)}{|a|} \sin 2|a| \\ \eta_{3}=\frac{\xi_{2}+\xi_{3}}{2}-\frac{\xi_{2}-\xi_{3}}{2} \cos 2|a|-\frac{\left(a, x_{1}\right)}{|a|} \sin 2|a|, \end{array}\right. \\ & \left\{\begin{array}{l} y_{1}=x_{1}-\frac{\left(\xi_{2}-\xi_{3}\right) a}{2|a|} \sin 2|a|-\frac{2\left(a, x_{1}\right) a}{|a|^{2}} \sin ^{2}|a| \\ y_{2}=x_{2} \cos |a|-\frac{\overline{x_{3} a}}{|a|} \sin |a| \\ y_{3}=x_{3} \cos |a|+\frac{\overline{a x}}{|a|} \sin |a| \end{array}\right. \end{aligned}

\alpha X=\left(\begin{array}{ccc} \xi_{1} & 0 & 0 \\ 0 & \xi_{2} & 0 \\ 0 & 0 & \xi_{3} \end{array}\right), \quad \xi_{i} \in \boldsymbol{R}

\alpha X = \pmatrix{\xi_1 & 0 & 0 \cr 0 & \xi_2 & 0 \cr 0 & 0 & \xi_3}, \quad \xi_i \in \text{$R$}.

\begin{aligned} & \eta_{1}(t)^{2}+\eta_{2}(t)^{2}+\eta_{3}(t)^{2} \\ & =\xi_{1}{ }^{2}+\left(\frac{\xi_{2}+\xi_{3}}{2}+\frac{\xi_{2}-\xi_{3}}{2} \cos 2 t+\left|x_{1}\right| \sin 2 t\right)^{2} \\ & \quad+\left(\frac{\xi_{2}+\xi_{3}}{2}-\frac{\xi_{2}-\xi_{3}}{2} \cos 2 t-\left|x_{1}\right| \sin 2 t\right)^{2} \\ & =\xi_{1}{ }^{2}+2\left(\frac{\xi_{2}+\xi_{3}}{2}\right)^{2}+2\left(\frac{\xi_{2}-\xi_{3}}{2} \cos 2 t+\left|x_{1}\right| \sin 2 t\right)^{2} \\ & =\xi_{1}{ }^{2}+2\left(\frac{\xi_{2}+\xi_{3}}{2}\right)^{2}+2\left(\left(\frac{\xi_{2}-\xi_{3}}{2}\right)^{2}+\left|x_{1}\right|^{2}\right) \sin ^{2}\left(2 t+t_{0}\right)\left(\text { for some } t_{0} \in \boldsymbol{R}\right) \\ & \leq{\xi_{1}}^{2}+2\left(\frac{\xi_{2}+\xi_{3}}{2}\right)^{2}+2\left(\left(\frac{\xi_{2}-\xi_{3}}{2}\right)^{2}+\left|x_{1}\right|^{2}\right) \\ & ={\xi_{1}}^{2}+{\xi_{2}}^{2}+{\xi_{3}}^{2}+2\left|x_{1}\right|^{2} \end{aligned}

\begin{aligned} & \operatorname{tr}(X)=\operatorname{tr}(\alpha X)(\text { Lemma 2.2.1.(2) })=\sum_{i=1}^{3} \xi_{i}, \\ & (X, X)=(\alpha X, \alpha X)(\text { Lemma 2.2.4 })=\sum_{i=1}^{3} \xi_{i}{ }^{2}, \\ & \operatorname{tr}(X, X, X)=\operatorname{tr}(\alpha X, \alpha X, \alpha X)(\text { Lemma 2.2.4 })=\sum_{i=1}^{3} \xi_{i}{ }^{3} . \end{aligned}

\left\{\begin{array}{l} \xi_{1}+\xi_{2}+\xi_{3}=\operatorname{tr}(X) \\ \xi_{1}{ }^{2}+\xi_{2}{ }^{2}+\xi_{3}{ }^{2}=(X, X) \\ \xi_{1}{ }^{3}+\xi_{2}{ }^{3}+\xi_{3}{ }^{3}=\operatorname{tr}(X, X, X) \end{array}\right.

\left\{\begin{array}{l} \xi_1 + \xi_2 + \xi_3 = \text\mathrm{{tr}}(X) \vspace{1mm}\\ {\xi_1}^2 + {\xi_2}^2 + {\xi_3}^2 = (X, X) \vspace{1mm}\\ {\xi_1}^3 + {\xi_2}^3 + {\xi_3}^3 = \text\mathrm{{tr}}(X, X, X). \end{array}\right.

\mathfrak{C} P_{2}=\left\{X \in \mathfrak{J} \mid X^{2}=X, \operatorname{tr}(X)=1\right\} .

\text{\es {C}} P_2 = \{ X \in \text{\es {J}} \, | \, X^2 = X, \text\mathrm{{tr}}(X) = 1 \}.

F_{4} / \operatorname{Spin}(9) \simeq \mathfrak{C} P_{2} .

\alpha X=\left(\begin{array}{ccc} \xi_{1} & 0 & 0 \\ 0 & \xi_{2} & 0 \\ 0 & 0 & \xi_{3} \end{array}\right), \quad \xi_{i} \in \boldsymbol{R}

\alpha X = \pmatrix{\xi_1 & 0 & 0 \cr 0 & \xi_2 & 0 \cr 0 & 0 & \xi_3}, \quad \xi_i \in \text{$R$}

\left(\begin{array}{ccc} \xi_{1}{ }^{2} & 0 & 0 \\ 0 & \xi_{2}{ }^{2} & 0 \\ 0 & 0 & \xi_{3}{ }^{2} \end{array}\right)=\left(\begin{array}{ccc} \xi_{1} & 0 & 0 \\ 0 & \xi_{2} & 0 \\ 0 & 0 & \xi_{3} \end{array}\right) .

\pmatrix{{\xi_1}^2 & 0 & 0 \cr 0 & {\xi_2}^2 & 0 \cr 0 & 0 & {\xi_3}^2 } = \pmatrix{\xi_1 & 0 & 0 \cr 0 & \xi_2 & 0 \cr 0 & 0 & \xi_3}.

\sigma\left(\begin{array}{ccc} \xi_{1} & x_{3} & \bar{x}_{2} \\ \bar{x}_{3} & \xi_{2} & x_{1} \\ x_{2} & \bar{x}_{1} & \xi_{3} \end{array}\right)=\left(\begin{array}{ccc} \xi_{1} & -x_{3} & -\bar{x}_{2} \\ -\bar{x}_{3} & \xi_{2} & x_{1} \\ -x_{2} & \bar{x}_{1} & \xi_{3} \end{array}\right) .

\sigma\pmatrix{\xi_1 & x_3 & \overline{x}_2 \cr \overline{x}_3 & \xi_2 & x_1 \cr x_2 & \overline{x}_1 & \xi_3} = \pmatrix{\xi_1 & -x_3 & -\overline{x}_2 \cr -\overline{x}_3 & \xi_2 & x_1 \cr -x_2 & \overline{x}_1 & \xi_3}.

\left(F_{4}\right)^{\sigma}=\left\{\alpha \in F_{4} \mid \sigma \alpha=\alpha \sigma\right\} .

(F_4)^{\sigma} = \{ \alpha \in F_4 \, | \, \sigma\alpha = \alpha\sigma \}.

\begin{aligned} \mathfrak{J}_{\sigma} & =\{X \in \mathfrak{J} \mid \sigma X=X\}=\left\{\left.\left(\begin{array}{ccc} \xi_{1} & 0 & 0 \\ 0 & \xi_{2} & x_{1} \\ 0 & \overline{x_{1}} & \xi_{3} \end{array}\right) \right\rvert\, \xi_{i} \in \boldsymbol{R}, x_{1} \in \mathfrak{C}\right\} \\ & \left.=\left\{X \in \mathfrak{J} \mid E_{1} \circ X=0\right\} \oplus \mathfrak{E}_{1} \text { (where } \mathfrak{E}_{1}=\left\{\xi E_{1} \mid \xi \in \boldsymbol{R}\right\}\right) \\ & =\mathfrak{J}(2, \mathfrak{C}) \oplus \mathfrak{E}_{1}, \\ \mathfrak{J}_{-\sigma} & =\{X \in \mathfrak{J} \mid \sigma X=-X\}=\left\{\left.\left(\begin{array}{ccc} 0 & x_{3} & \bar{x}_{2} \\ \bar{x}_{3} & 0 & 0 \\ x_{2} & 0 & 0 \end{array}\right) \right\rvert\, x_{i} \in \mathfrak{C}\right\} \\ & =\left\{X \in \mathfrak{J} \mid 2 E_{1} \circ X=X\right\} . \end{aligned}

\alpha E_{2}, \alpha E_{3} \in \mathfrak{J}(2, \mathfrak{C}) .

\alpha E_2, \alpha E_3 \in \text{\es {J}}(2, \text{\es {C}}).

\begin{aligned} \alpha E_{2} & =\alpha\left(-F_{2}(1) \times F_{2}(1)\right)=-\alpha F_{2}(1) \times \alpha F_{2}(1) \\ & =-\left(F_{2}\left(x_{2}\right)+F_{3}\left(x_{3}\right)\right) \times\left(F_{2}\left(x_{2}\right)+F_{3}\left(x_{3}\right)\right) \quad\left(\text { for some } x_{2}, x_{3} \in \mathfrak{C}\right) \\ & =\left(x_{2}, x_{2}\right) E_{2}+\left(x_{3}, x_{3}\right) E_{3}-F_{1}\left(\overline{x_{2} x_{3}}\right) \in \mathfrak{J}(2, \mathfrak{C}) \end{aligned}

\begin{aligned} \alpha E_2 \!\!\! &=& \!\!\! \alpha(-F_2(1) \times F_2(1)) = -\alpha F_2(1) \times \alpha F_2(1) \\ \!\!\! &=& \!\!\! -(F_2(x_2) + F_3(x_3)) \times (F_2(x_2) + F_3(x_3)) \;\;(\text{for some } x_2, x_3 \in \text{\es {C}}) \\ \!\!\! &=& \!\!\! (x_2, x_2)E_2 + (x_3, x_3)E_3 - F_1(\overline{x_2x_3})\in \text{\es {J}}(2,\text{\es {C}}). \end{aligned}

\alpha E_{1}=E_{1}+\xi_{2} E_{2}+\xi_{3} E_{3}+F_{1}\left(x_{1}\right) .

\alpha E_1 = E_1 + \xi_2E_2 + \xi_3E_3 + F_1(x_1).

1=\left(E_{1}, E_{1}\right)=\left(\alpha E_{1}, \alpha E_{1}\right)=1+\xi_{2}{ }^{2}+\xi_{3}{ }^{2}+2\left|x_{1}\right|^{2},

1 = (E_1, E_1) = (\alpha E_1, \alpha E_1) = 1 + {\xi_2}^2 + {\xi_3}^2 + 2|x_1|^2,

\begin{aligned} \alpha \sigma X & =\alpha \sigma\left(X_{1}+X_{2}\right) \quad X_{1} \in \mathfrak{J}_{\sigma}, \quad X_{2} \in \mathfrak{J}_{-\sigma} \\ & =\alpha\left(X_{1}-X_{2}\right)=\alpha X_{1}-\alpha X_{2}=\sigma\left(\alpha X_{1}\right)+\sigma\left(\alpha X_{2}\right) \\ & =\sigma \alpha\left(X_{1}+X_{2}\right)=\sigma \alpha X, \quad X \in \mathfrak{J} \end{aligned}

\begin{aligned} \alpha\sigma X \!\!\! &=& \!\!\! \alpha\sigma(X_1 + X_2) \qquad X_1 \in \text{\es {J}}_{\sigma},\;\; X_2 \in \text{\es {J}}_{-\sigma} \\ \!\!\! &=& \!\!\! \alpha(X_1 - X_2) = \alpha X_1 - \alpha X_2 = \sigma(\alpha X_1) + \sigma(\alpha X_2) \\ \!\!\! &=& \!\!\! \sigma\alpha(X_1 + X_2) = \sigma\alpha X, \quad X \in \text{\es {J}}. \end{aligned}

z\left(F_{4}\right)=\{1\} .

z(F_4) = \{ 1 \}.

\alpha=1 \quad \text { or } \quad \alpha=\sigma .

\alpha = 1 \quad \text{ or } \quad \alpha = \sigma.

\gamma\left(\begin{array}{ccc} \xi_{1} & x_{3} & \bar{x}_{2} \\ \bar{x}_{3} & \xi_{2} & x_{1} \\ x_{2} & \bar{x}_{1} & \xi_{3} \end{array}\right)=\left(\begin{array}{ccc} \xi_{1} & \gamma x_{3} & \overline{\gamma x_{2}} \\ \overline{\gamma x_{3}} & \xi_{2} & \gamma x_{1} \\ \gamma x_{2} & \overline{\gamma x_{1}} & \xi_{3} \end{array}\right) .

\gamma\pmatrix{\xi_1 & x_3 & \overline{x}_2 \cr \overline{x}_3 & \xi_2 & x_1 \cr x_2 & \overline{x}_1 & \xi_3} = \pmatrix{\xi_1 & \gamma x_3 & \overline{\gamma x_2} \cr \overline{\gamma x_3} & \xi_2 & \gamma x_1 \cr \gamma x_2 & \overline{\gamma x_1} & \xi_3}.

\left(F_{4}\right)^{\gamma}=\left\{\alpha \in F_{4} \mid \gamma \alpha=\alpha \gamma\right\} .

(F_4)^{\gamma} = \{ \alpha \in F_4 \, | \, \gamma\alpha = \alpha\gamma \}.

X=\left(\begin{array}{ccc} \xi_{1} & x_{3} & \bar{x}_{2} \\ \bar{x}_{3} & \xi_{2} & x_{1} \\ x_{2} & \bar{x}_{1} & \xi_{3} \end{array}\right)=\left(\begin{array}{ccc} \xi_{1} & m_{3} & \bar{m}_{2} \\ \bar{m}_{3} & \xi_{2} & m_{1} \\ m_{2} & \bar{m}_{1} & \xi_{3} \end{array}\right)+\left(\begin{array}{ccc} 0 & a_{3} e_{4} & -a_{2} e_{4} \\ -a_{3} e_{4} & 0 & a_{1} e_{4} \\ a_{2} e_{4} & -a_{1} e_{4} & 0 \end{array}\right),

\left(\begin{array}{ccc} \xi_{1} & m_{3} & \bar{m}_{2} \\ \bar{m}_{3} & \xi_{2} & m_{1} \\ m_{2} & \bar{m}_{1} & \xi_{3} \end{array}\right)+\left(a_{1}, a_{2}, a_{3}\right) .

\pmatrix{\xi_1 & m_3 & \overline{m}_2 \cr \overline{m}_3 & \xi_2 & m_1 \cr m_2 & \overline{m}_1 & \xi_3} + (a_1, a_2, a_3).

\begin{aligned} (M+\boldsymbol{a}) \times(N+\boldsymbol{b}) & =\left(M \times N-\frac{1}{2}\left(\boldsymbol{a}^{*} \boldsymbol{b}+\boldsymbol{b}^{*} \boldsymbol{a}\right)\right)-\frac{1}{2}(\boldsymbol{a} N+\boldsymbol{b} M), \\ (M+\boldsymbol{a}, N+\boldsymbol{b}) & =(M, N)+2(\boldsymbol{a}, \boldsymbol{b}), \\ \gamma(M+\boldsymbol{a}) & =M-\boldsymbol{a} . \end{aligned}

\begin{aligned} (M + \text{$a$}) \times (N + \text{$b$}) \!\!\! &=& \!\!\! \Big(M \times N - \frac{1}{2} (\text{$a$}^*\text{$b$} + \text{$b$}^*\text{$a$})\Big) - \frac{1}{2}(\text{$a$} N + \text{$b$} M), \\ (M + \text{$a$}, N + \text{$b$}) \!\!\! &=& \!\!\! (M, N) + 2(\text{$a$}, \text{$b$}), \\ \gamma(M + \text{$a$}) \!\!\! &=& \!\!\! M - \text{$a$}. \end{aligned}

\mathfrak{J}(3, \boldsymbol{H}) \oplus \boldsymbol{H}^{3}=\mathfrak{J} .

\text{\es {J}}(3, \text{$H$}) \oplus \text{$H$}^3 = \text{\es {J}}.

\begin{aligned} F_{4, \boldsymbol{H}} & =\left\{\alpha \in \operatorname{Iso}_{\boldsymbol{R}}\left(\mathfrak{J}_{\boldsymbol{H}}\right) \mid \alpha(M \circ N)=\alpha M \circ \alpha N\right\} \\ & =\left\{\alpha \in \operatorname{Iso}_{\boldsymbol{R}}\left(\mathfrak{J}_{\boldsymbol{H}}\right) \mid \alpha(M \times N)=\alpha M \times \alpha N\right\} . \end{aligned}

\begin{aligned} F_{4,{\text{${H}$}}} \!\!\! &=& \!\!\! \{ \alpha \in \text\mathrm{{Iso}}_{\text{${R}$}}(\text{\es {J}}_{\text{${H}$}}) \, | \, \alpha(M \circ N) = \alpha M \circ \alpha N \} \\ \!\!\! &=& \!\!\! \{ \alpha \in \text\mathrm{{Iso}}_{\text{${R}$}}(\text{\es {J}}_{\text{${H}$}}) \, | \, \alpha(M \times N) = \alpha M \times \alpha N \}. \end{aligned}

\varphi(A) M=A M A^{*}, \quad M \in \mathfrak{J}_{\boldsymbol{H}} .

\varphi(A)M = AMA^*, \quad M \in \text{\es {J}}_{\text{${H}$}}.

\alpha E_{i}=A E_{i} A^{*}, \quad i=1,2,3 .

\alpha E_i = AE_iA^*, \quad i = 1, 2, 3.

\beta E_{i}=E_{i}, \quad i=1,2,3 .

\beta E_i = E_i, \quad i = 1, 2, 3.

\left(\beta_{1} m\right)\left(\beta_{2} n\right)=\overline{\beta_{3}(\overline{m n})}, \quad m, n \in \boldsymbol{H} .

\displaylines{\hfill (\beta_1m)(\beta_2n) = \overline{\beta_3(\overline{mn})}, \quad m, n \in \text{$H$}. \hfill\text{(i)}}

\beta_{2} m=\bar{p}\left(\beta_{1} m\right) q, \quad \beta_{3} m=\overline{\left(\beta_{1} \bar{m}\right)} q .

\beta_2m = \overline{p}(\beta_1m)q, \quad \beta_3m = \overline{(\beta_1\overline{m})}q.

(\zeta m)(\zeta n)=\zeta(m n), \quad m, n \in \boldsymbol{H},

(\zeta m)(\zeta n) = \zeta(mn), \quad m, n \in \text{$H$},

\beta_{1} m=\operatorname{prm} \bar{r}, \quad \beta_{2} m=r m \bar{r} q, \quad \beta_{3} m=\bar{q} r m \bar{r} \bar{p}, \quad m \in \boldsymbol{H} .

\beta_1m = prm\overline{r}, \;\; \beta_2m = rm\overline{r}q, \;\; \beta_3m = \overline{q}rm \overline{r}\,\overline{p}, \quad m \in \text{$H$}.

\beta M=B M B^{*}, \quad M \in \mathfrak{J}_{\boldsymbol{H}},

\beta M = BMB^*, \quad M \in \text{\es {J}}_{\text{${H}$}},

\alpha=\varphi(A) \beta=\varphi(A) \varphi(B)=\varphi(A B), \quad A B \in \operatorname{Sp}(3) .

\alpha = \varphi(A)\beta = \varphi(A)\varphi(B) = \varphi(AB), \quad AB \in Sp(3).

\varphi(p, A)(M+\boldsymbol{a})=A M A^{*}+p \boldsymbol{a} A^{*}, \quad M+\boldsymbol{a} \in \mathfrak{J}_{\boldsymbol{H}} \oplus \boldsymbol{H}^{3}=\mathfrak{J} .

\varphi(p,A)(M + \text{$a$}) = AMA^* + p\text{$a$} A^*, \quad M + \text{$a$} \in \text{\es {J}}_{\text{${H}$}} \oplus \text{$H$}^3 = \text{\es {J}}.

\begin{aligned} A M A^{*} \times A N A^{*} & =A(M \times N) A^{*} \\ \left(p \boldsymbol{a} A^{*}\right)^{*}\left(p \boldsymbol{b} A^{*}\right) & =A \boldsymbol{a}^{*} \bar{p} p \boldsymbol{b} A^{*}=A\left(\boldsymbol{a}^{*} \boldsymbol{b}\right) A^{*} \\ \left(p \boldsymbol{a} A^{*}\right)\left(A N A^{*}\right) & =p(\boldsymbol{a} N) A^{*}, \text { etc. } \end{aligned}

\begin{aligned} AMA^* \times ANA^* \!\!\!&=&\!\!\! A(M \times N)A^*, \vspace{1mm}\\ (p\text{$a$} A^*)^*(p\text{$b$} A^*) \!\!\!&=&\!\!\! A\text{$a$}^*\overline{p}p\text{$b$} A^* = A(\text{$a$}^*\text{$b$})A^*, \vspace{1mm}\\ (p\text{$a$} A^*)(ANA^*) \!\!\!&=&\!\!\! p(\text{$a$} N)A^*,\;\;\text{etc.} \end{aligned}

\varphi(p, A)((M+\boldsymbol{a}) \times(N+\boldsymbol{b}))=\varphi(p, A)(M+\boldsymbol{a}) \times \varphi(p, A)(N+\boldsymbol{b}),

\varphi(p, A)((M + \text{$a$}) \times (N + \text{$b$})) = \varphi(p, A)(M + \text{$a$}) \times \varphi(p, A)(N + \text{$b$}),

\alpha M=A M A^{*}, \quad M \in \mathfrak{J}_{\boldsymbol{H}}

\alpha M = AMA^*, \quad M \in \text{\es {J}}_{\text{${H}$}}

\beta\left(m+a e_{4}\right)=m+(p a) e_{4}, \quad m+a e_{4} \in \boldsymbol{H} \oplus \boldsymbol{H} e_{4}=\mathfrak{C},

\beta(m + ae_4) = m + (pa)e_4, \quad m + ae_4 \in \text{$H$} \oplus \text{$H$} e_4 =\text{\es {C}},

\beta(M+\boldsymbol{a})=M+p \boldsymbol{a}=\varphi(p, E)(M+\boldsymbol{a}),

\beta(M + \text{$a$}) = M + p\text{$a$} = \varphi(p, E)(M + \text{$a$}),

\begin{aligned} & G_{i j}, \quad 0 \leq i<j \leq 3,4 \leq i<j \leq 7, \\ & \widetilde{A}_{k}\left(e_{j}\right), \quad 0 \leq j \leq 3, k=1,2,3 \end{aligned}

\begin{array}{l} G_{ij}, \quad 0 \le i < j \le 3, 4 \le i < j \le 7, \vspace{1mm}\\ \widetilde{A}_k(e_j), \quad 0 \le j \le 3, k = 1, 2, 3 \end{array}

w\left(\begin{array}{ccc} \xi_{1} & x_{3} & \bar{x}_{2} \\ \bar{x}_{3} & \xi_{2} & x_{1} \\ x_{2} & \bar{x}_{1} & \xi_{3} \end{array}\right)=\left(\begin{array}{ccc} \xi_{1} & \omega x_{3} & \overline{\omega x_{2}} \\ \overline{\omega x_{3}} & \xi_{2} & \omega x_{1} \\ \omega x_{2} & \overline{\omega x_{1}} & \xi_{3} \end{array}\right) .

w\pmatrix{\xi_1 & x_3 & \overline{x}_2 \cr \overline{x}_3 & \xi_2 & x_1 \cr x_2 & \overline{x}_1 & \xi_3} = \pmatrix{\xi_1 & \omega x_3 & \overline{\omega x_2} \cr \overline{\omega x_3} & \xi_2 & \omega x_1 \cr \omega x_2 & \overline{\omega x_1} & \xi_3}.

\left(F_{4}\right)^{w}=\left\{\alpha \in F_{4} \mid w \alpha=\alpha w\right\} .

(F_4)^w = \{\alpha \in F_4 \, |\, w\alpha = \alpha w \}.

\left(\begin{array}{ccc} \xi_{1} & x_{3} & \bar{x}_{2} \\ \bar{x}_{3} & \xi_{2} & x_{1} \\ x_{2} & \bar{x}_{1} & \xi_{3} \end{array}\right), \quad \xi_{i} \in \boldsymbol{R}, x_{i}=a_{i}+\boldsymbol{m}_{i} \in \boldsymbol{C} \oplus \boldsymbol{C}^{3}=\mathfrak{C}

\pmatrix{\xi_1 & x_3 & \overline{x}_2 \cr \overline{x}_3 & \xi_2 & x_1 \cr x_2 & \overline{x}_1 & \xi_3}, \quad \xi_i \in \text{$R$}, \, x_i = a_i + \text{$m$}_i \in \text{$C$} \oplus \text{$C$}^3 = \text{\es {C}}

\left(\begin{array}{lll} \xi_{1} & a_{3} & \bar{a}_{2} \\ \bar{a}_{3} & \xi_{2} & a_{1} \\ a_{2} & \bar{a}_{1} & \xi_{3} \end{array}\right)+\left(\boldsymbol{m}_{1}, \boldsymbol{m}_{2}, \boldsymbol{m}_{3}\right)

\pmatrix{\xi_1 & a_3 & \overline{a}_2 \cr \overline{a}_3 & \xi_2 & a_1 \cr a_2 & \overline{a}_1 & \xi_3} + \Big(\text{$m$}_1, \text{$m$}_2, \text{$m$}_3 \Big)

M \times N=\left(\begin{array}{ccc} \boldsymbol{m}_{2} \times \boldsymbol{n}_{3} & \boldsymbol{m}_{3} \times \boldsymbol{n}_{1} & \boldsymbol{m}_{1} \times \boldsymbol{n}_{2} \\ + & + & + \\ \boldsymbol{n}_{2} \times \boldsymbol{m}_{3} & \boldsymbol{n}_{3} \times \boldsymbol{m}_{1} & \boldsymbol{n}_{1} \times \boldsymbol{m}_{2} \end{array}\right) \in M(3, C),

M \times N = \pmatrix{\text{$m$}_2 \times \text{$n$}_3 & \text{$m$}_3 \times \text{$n$}_1 & \text{$m$}_1 \times \text{$n$}_2 \cr + & + & + \cr \text{$n$}_2 \times \text{$m$}_3 & \text{$n$}_3 \times \text{$m$}_1 & \text{$n$}_1 \times \text{$m$}_2} \in M(3, \text{$C$}),

P M \times P N={ }^{t} \widetilde{P}(M \times N), \quad M A \times N A=(M \times N)^{t} \widetilde{A},

PM \times PN = {}^t\widetilde{P}(M \times N), \quad MA \times NA = (M \times N)\,{}^t\!\widetilde{A},

(M, N)=\frac{1}{2} \operatorname{tr}\left(M^{*} N+N^{*} M\right)=\sum_{i, j}\left(m_{i j}, n_{i j}\right),

(M, N) = \frac{1}{2}\text\mathrm{{tr}}(M^*N + N^*M) = \sum_{i,j}(m_{ij}, n_{ij}),

\begin{aligned} (X+M) \times(Y+N) & =\left(X \times Y-\frac{1}{2}\left(M^{*} N+N^{*} M\right)\right)-\frac{1}{2}(M Y+N X+\overline{M \times N}), \\ (X+M, Y+N) & =(X, Y)+2(M, N), \\ w(X+M) & =X+\omega_{1} M, \quad\left(\omega_{1}=-\frac{1}{2}+\frac{\sqrt{3}}{2} e_{1}\right), \end{aligned}

\begin{aligned} (X + M) \times (Y + N) \!\!\! &=& \!\!\! (X \times Y - \frac{1}{2}(M^*N + N^*M)) - \frac{1}{2}(MY + NX + \overline{M \times N}), \vspace{1mm}\\ (X + M, Y + N) \!\!\! &=& \!\!\! (X, Y) + 2(M, N), \vspace{1mm}\\ w(X + M) \!\!\! &=& \!\!\! X + \omega_1M, \quad \Big(\omega_1 = - \frac{1}{2} + \frac{\sqrt{3}}{2}e_1\Big), \end{aligned}

\mathfrak{J}(3, \boldsymbol{C}) \oplus M(3, \boldsymbol{C})=\mathfrak{J}

\text{\es {J}}(3, \text{$C$}) \oplus M(3, \text{$C$}) = \text{\es {J}}.

\begin{aligned} F_{4, \boldsymbol{C}} & =\left\{\alpha \in \operatorname{Iso}_{\boldsymbol{R}}\left(\mathfrak{J}_{\boldsymbol{C}}\right) \mid \alpha(X \circ Y)=\alpha X \circ \alpha Y\right\} \\ & =\left\{\alpha \in \operatorname{Iso}_{\boldsymbol{R}}\left(\mathfrak{J}_{\boldsymbol{C}}\right) \mid \alpha(X \times Y)=\alpha X \times \alpha Y\right\} . \end{aligned}

\begin{aligned} F_{4,\text{${C}$}} \!\!\! &=& \!\!\! \{\alpha \in \text\mathrm{{Iso}}_{\text{${R}$}}(\text{\es {J}}_{\text{${C}$}}) \, | \, \alpha(X \circ Y) = \alpha X \circ \alpha Y \} \vspace{1mm}\\ \!\!\! &=& \!\!\! \{\alpha \in \text\mathrm{{Iso}}_{\text{${R}$}}(\text{\es {J}}_{\text{${C}$}}) \, | \, \alpha(X \times Y) = \alpha X \times \alpha Y \}. \end{aligned}

\epsilon A=\bar{A}, \quad A \in S U(3),

\epsilon A = \overline{A}, \quad A \in SU(3),

\varphi(A, 1) X=A X A^{*}, \quad \varphi(A, \epsilon) X=A \bar{X} A^{*}, \quad X \in \mathfrak{J}_{\boldsymbol{C}} .

\varphi(A, 1)X = AXA^*, \quad \varphi(A, \epsilon)X = A\overline{X}A^*, \quad X \in \text{\es {J}}_{\text{${C}$}}.

\beta_{1} x=p x, \beta_{2} x=x q, \beta_{3} x=\bar{q} x \bar{p} \quad \text { or } \quad \beta_{1} x=p \bar{x}, \beta_{2} x=\bar{x} q, \beta_{3} x=\bar{q} \bar{x} \bar{p} .

\beta_1x = px, \beta_2x = xq, \beta_3x = \overline{q}x\overline{p} \quad \text{or} \quad \beta_1x = p\overline{x}, \beta_2x = \overline{x}q, \beta_3x = \overline{q}\,\overline{x}\,\overline{p}.

\beta X=B X B^{*} \quad \text { or } \quad \beta X=B \bar{X} B^{*}, \quad X \in \mathfrak{J}_{C} .

\beta X = BXB^* \quad \text{or} \quad \beta X = B\overline{X}B^*, \quad X \in \text{\es {J}}_{\text{${C}$}}.

\varphi(P, A)(X+M)=A X A^{*}+P M A^{*}, \quad X+M \in \mathfrak{J}(3, \boldsymbol{C}) \oplus M(3, \boldsymbol{C})=\mathfrak{J} .

\varphi(P, A)(X + M) = AXA^* + PMA^*, \quad X + M \in \text{\es {J}}(3, \text{$C$}) \oplus M(3, \text{$C$}) = \text{\es {J}}.

\begin{aligned} & A X A^{*} \times A Y A^{*}=A(X \times Y) A^{*} \\ & \left(P M A^{*}\right)^{*}\left(P N A^{*}\right)=A M^{*} P^{*} P N A=A\left(M^{*} N\right) A, \\ & \left(P M A^{*}\right)\left(A Y A^{*}\right)=P(M Y) A^{*}, \\ & \overline{P M A^{*} \times P N A^{*}}=\frac{\bar{t} \widetilde{P}(M \times N)^{t} \widetilde{A} *}{}=P \overline{M \times N} A^{*}, \text { etc. } \end{aligned}

\varphi(P, A)((X+M) \times(Y+N))=\varphi(P, A)(X+M) \times \varphi(P, A)(Y+N),

\varphi(P, A)((X + M) \times (Y + N)) = \varphi(P, A)(X + M) \times \varphi(P, A)(Y + N),

\alpha X=A X A^{*} \quad \text { or } \quad \alpha X=A \bar{X} A^{*}, \quad X \in \mathfrak{J}_{C}

\alpha X = AXA^* \quad \text{or} \quad \alpha X = A\overline{X}A^*, \quad X \in \text{\es {J}}_{\text{${C}$}}

\beta(X+M)=X+P M=\varphi(P, E)(X+M), \quad X+M \in \mathfrak{J}_{C} \oplus M(3, \boldsymbol{C})=\mathfrak{J} .

\beta(X + M) = X + PM = \varphi(P, E)(X + M), \quad X + M \in \text{\es {J}}_C \oplus M(3, \text{$C$}) = \text{\es {J}}.

\alpha=\varphi(E, A) \beta=\varphi(E, A) \varphi(P, E)=\varphi(P, A) .

\alpha = \varphi(E, A)\beta = \varphi(E, A)\varphi(P, E) = \varphi(P, A).

\begin{aligned} & G_{01}, \quad G_{23}, \quad G_{45}, \quad G_{67}, \quad G_{26}+G_{37}, \quad-G_{27}+G_{36}, \\ & G_{24}+G_{35}, \quad-G_{25}+G_{34}, \quad G_{46}+G_{57}, \quad-G_{47}+G_{56}, \\ & \widetilde{A}_{1}(1), \quad \widetilde{A}_{2}(1), \quad \widetilde{A}_{3}(1), \quad \widetilde{A}_{1}\left(e_{1}\right), \quad \widetilde{A}_{2}\left(e_{1}\right), \quad \widetilde{A}_{3}\left(e_{1}\right) \end{aligned}

\begin{aligned} F_{4}^{C} & =\left\{\alpha \in \operatorname{Iso}_{C}\left(\mathfrak{J}^{C}\right) \mid \alpha(X \circ Y)=\alpha X \circ \alpha Y\right\} \\ & =\left\{\alpha \in \operatorname{Iso}_{C}\left(\mathfrak{J}^{C}\right) \mid \operatorname{det}(\alpha X)=\operatorname{det} X,(\alpha X, \alpha Y)=(X, Y)\right\} \\ & =\left\{\alpha \in \operatorname{Iso}_{C}\left(\mathfrak{J}^{C}\right) \mid \operatorname{det}(\alpha X)=\operatorname{det} X, \alpha E=E\right\} \\ & =\left\{\alpha \in \operatorname{Iso}_{C}\left(\mathfrak{J}^{C}\right) \mid \alpha(X \times Y)=\alpha X \times \alpha Y\right\} . \end{aligned}

\langle X, Y\rangle=(\tau X, Y) .

\langle X, Y \rangle = (\tau X, Y).

F_{4}=\left\{\alpha \in F_{4}{ }^{C} \mid \tau \alpha=\alpha \tau\right\} .

F_4 = \{ \alpha \in {F_4}^C \, | \, \tau\alpha = \alpha\tau \}.

F_{4}{ }^{C} \simeq F_{4} \times \boldsymbol{R}^{52} .

{F_4}^C \simeq F_4 \times \text{$R$}^{52}.

F_{4}{ }^{C} \simeq\left(F_{4}{ }^{C} \cap U\left(\mathfrak{J}^{C}\right)\right) \times \boldsymbol{R}^{d}=F_{4} \times \boldsymbol{R}^{d},

{F_4}^C \simeq ({F_4}^C \cap U(\text{\es {J}}^C)) \times \text{$R$}^d = F_4 \times \text{$R$}^d,

\begin{aligned} \mathfrak{J}\left(3, \mathfrak{C}^{\prime}\right) & =\left\{X \in M\left(3, \mathfrak{C}^{\prime}\right) \mid X^{*}=X\right\}, \\ \mathfrak{J}(1,2, \mathfrak{C}) & =\left\{X \in M(3, \mathfrak{C}) \mid I_{1} X^{*} I_{1}=X\right\}, \end{aligned}

\begin{aligned} \text{\es {J}}(3, \text{\es {C}}') \!\!\! &=& \!\!\! \{ X \in M(3, \text{\es {C}}') \, | \, X^* = X \}, \\ \text{\es {J}}(1, 2, \text{\es {C}}) \!\!\! &=& \!\!\! \{ X \in M(3, \text{\es {C}}) \, | \, I_1X^*I_1 = X \}, \end{aligned}

X \circ Y=\frac{1}{2}(X Y+Y X)

X \circ Y = \frac{1}{2}(XY + YX).

\begin{aligned} \mathfrak{J}\left(3, \mathfrak{C}^{\prime}\right) & \cong\left\{X \in \mathfrak{J}\left(3, \mathfrak{C}^{C}\right) \mid \tau \gamma X=X\right\}=\left(\mathfrak{J}\left(3, \mathfrak{C}^{C}\right)\right)_{\tau \gamma}, \\ \mathfrak{J}(1,2, \mathfrak{C}) & \cong\left\{X \in \mathfrak{J}\left(3, \mathfrak{C}^{C}\right) \mid \tau \sigma X=X\right\}=\left(\mathfrak{J}\left(3, \mathfrak{C}^{C}\right)\right)_{\tau \sigma} \end{aligned}

\begin{aligned} F_{4(4)} & =\left\{\alpha \in \operatorname{Iso}_{\boldsymbol{R}}\left(\mathfrak{J}\left(3, \mathfrak{C}^{\prime}\right)\right) \mid \alpha(X \circ Y)=\alpha X \circ \alpha Y\right\}, \\ F_{4(-20)} & =\left\{\alpha \in \operatorname{Iso}_{\boldsymbol{R}}(\mathfrak{J}(1,2, \mathfrak{C})) \mid \alpha(X \circ Y)=\alpha X \circ \alpha Y\right\} . \end{aligned}

\begin{aligned} F_{4(4)} \!\!\! &=& \!\!\! \{\alpha \in \text\mathrm{{Iso}}_{\text{${R}$}}(\text{\es {J}}(3, \text{\es {C}}')) \, | \, \alpha(X \circ Y) = \alpha X \circ \alpha Y \}, \vspace{1mm}\\ F_{4(-20)} \!\!\! &=& \!\!\! \{\alpha \in \text\mathrm{{Iso}}_{\text{${R}$}}(\text{\es {J}}(1, 2, \text{\es {C}})) \, | \, \alpha(X \circ Y) = \alpha X \circ \alpha Y \}. \end{aligned}

F_{4(4)}=\left(F_{4}{ }^{C}\right)^{\tau \gamma}, \quad F_{4(-20)}=\left(F_{4}{ }^{C}\right)^{\tau \sigma} .

F_{4(4)} = ({F_4}^C)^{\tau\gamma}, \quad F_{4(-20)} = ({F_4}^C)^{\tau\sigma}.

\begin{aligned} F_{4(4)} & \simeq(\operatorname{Sp}(1) \times \operatorname{Sp}(3)) / \boldsymbol{Z}_{2} \times \boldsymbol{R}^{28}, \\ F_{4(-20)} & \simeq \operatorname{Spin}(9) \times \boldsymbol{R}^{16} . \end{aligned}

\begin{aligned} F_{4(4)} \!\!\! &\simeq& \!\!\! (Sp(1) \times Sp(3))/\text{$Z$}_2 \times \text{$R$}^{28},\\ F_{4(-20)} \!\!\! &\simeq& \!\!\! Spin(9) \times \text{$R$}^{16}. \end{aligned}

z\left(F_{4(4)}\right)=\{1\}, \quad z\left(F_{4(-20)}\right)=\{1\} .

z(F_{4(4)}) = \{1\}, \quad z(F_{4(-20)}) = \{1\}.

\begin{aligned} E_{6}^{C} & =\left\{\alpha \in \operatorname{Iso}_{C}\left(\mathfrak{J}^{C}\right) \mid \operatorname{det}(\alpha X)=\operatorname{det} X\right\} \\ & =\left\{\alpha \in \operatorname{Iso}_{C}\left(\mathfrak{J}^{C}\right) \mid(\alpha X, \alpha Y, \alpha Z)=(X, Y, Z)\right\}, \\ E_{6} & =\left\{\alpha \in \operatorname{Iso}_{C}\left(\mathfrak{J}^{C}\right) \mid \operatorname{det}(\alpha X)=\operatorname{det} X,\langle\alpha X, \alpha Y\rangle=\langle X, Y\rangle\right\} \\ & =\left\{\alpha \in \operatorname{Iso}_{C}\left(\mathfrak{J}^{C}\right) \mid(\alpha X, \alpha Y, \alpha Z)=(X, Y, Z),\langle\alpha X, \alpha Y\rangle=\langle X, Y\rangle\right\} \\ & =\left\{\alpha \in \operatorname{Iso}_{C}\left(\mathfrak{J}^{C}\right) \mid \alpha X \times \alpha Y={ }^{t} \alpha^{-1}(X \times Y),\langle\alpha X, \alpha Y\rangle=\langle X, Y\rangle\right\} \\ & =\left\{\alpha \in \operatorname{Iso}_{C}\left(\mathfrak{J}^{C}\right) \mid \alpha X \times \alpha Y=\tau \alpha \tau(X \times Y),\langle\alpha X, \alpha Y\rangle=\langle X, Y\rangle\right\} . \end{aligned}

\begin{aligned} {E_6}^C \!\!\! &=& \!\!\! \{ \alpha \in \text\mathrm{{Iso}}_C(\text{\es {J}}^C) \, | \, \text\mathrm{{det}}\,(\alpha X) = \text\mathrm{{det}}\,X \} \\ \!\!\! &=& \!\!\! \{ \alpha \in \text\mathrm{{Iso}}_C(\text{\es {J}}^C) \, | \, (\alpha X, \alpha Y, \alpha Z) = (X, Y, Z) \}, \vspace{1mm}\\ E_6 \!\!\! &=& \!\!\! \{ \alpha \in \text\mathrm{{Iso}}_C(\text{\es {J}}^C) \, | \, \text\mathrm{{det}}\,(\alpha X) = \text\mathrm{{det}}\,X, \langle \alpha X, \alpha Y \rangle = \langle X, Y \rangle \} \\ \!\!\! &=& \!\!\! \{ \alpha \in \text\mathrm{{Iso}}_C(\text{\es {J}}^C) \, | \, (\alpha X, \alpha Y, \alpha Z) = (X, Y, Z), \langle \alpha X, \alpha Y \rangle = \langle X, Y \rangle \} \\ \!\!\! &=& \!\!\! \{ \alpha \in \text{\rm {Iso}}_C(\text{\es {J}}^C) \, | \, \alpha X \times \alpha Y = {}^t\alpha^{-1}(X \times Y), \langle \alpha X, \alpha Y \rangle = \langle X, Y \rangle \} \\ \!\!\! &=& \!\!\! \{ \alpha \in \text{\rm {Iso}}_C(\text{\es {J}}^C) \, | \, \alpha X \times \alpha Y = \tau\alpha\tau(X \times Y), \langle \alpha X, \alpha Y \rangle =\ \langle X, Y \rangle \}. \end{aligned}

\lambda(\alpha)={ }^{t} \alpha^{-1}, \quad \alpha \in E_{6}{ }^{C}

\lambda(\alpha) = {}^t\alpha^{-1}, \quad \alpha \in {E_6}^C

E_{6}=\left\{\alpha \in E_{6}{ }^{C} \mid \tau \lambda(\alpha) \tau=\alpha\right\}=\left(E_{6}{ }^{C}\right)^{\tau \lambda} .

E_6 = \{ \alpha \in {E_6}^C \, | \, \tau\lambda(\alpha)\tau = \alpha \} = ({E_6}^C)^{\tau\lambda}.

U(27)=U\left(\mathfrak{J}^{C}\right)=\left\{\alpha \in \operatorname{Iso}_{C}\left(\mathfrak{J}^{C}\right) \mid\langle\alpha X, \alpha Y\rangle=\langle X, Y\rangle\right\} .

U(27) = U(\text{\es {J}}^C) = \{ \alpha \in \text\mathrm{{Iso}}_C(\text{\es {J}}^C) \, | \, \langle \alpha X, \alpha Y \rangle = \langle X, Y \rangle \}.

\mathfrak{e}_{6}{ }^{C}=\left\{\phi \in \operatorname{Hom}_{C}\left(\mathfrak{J}^{C}\right) \mid(\phi X, X, X)=0\right\} .

{\text{\es {e}}_6}^C = \{ \phi \in \text\mathrm{{Hom}}_C (\text{\es {J}}^C) \, | \, (\phi X, X, X) = 0 \}.

\phi=\delta+\widetilde{T}, \quad \delta \in \mathfrak{f}_{4}^{C}, T \in \mathfrak{J}_{0}^{C} .

\phi = \delta + \widetilde{T}, \quad \delta \in {\text{\es {f}}_4}^C,T \in {\text{\es {J}}_0}^C.

\[ \operatorname{dim}_{C}\left(\mathfrak{e}_{6}{ }^{C}\right)=52+26=78 . \] \end{itemize}

\dim_C({\text{\es {e}}_6}^C) = 52 + 26 = 78.

\delta+\widetilde{T}=0, \quad \delta \in \mathfrak{f}_{4}{ }^{C}, T \in \mathfrak{J}_{0}{ }^{C} \quad \text { implies } \quad \delta=0, T=0 .

\delta + \widetilde{T} = 0,\;\; \delta \in {\text{\es {f}}_4}^C,T \in {\text{\es {J}}_0}^C \quad \text{implies} \quad \delta = 0,T = 0.

\left[\delta_{1}+\widetilde{T}_{1}, \delta_{2}+\widetilde{T}_{2}\right]=\left(\left[\delta_{1}, \delta_{2}\right]+\left[\widetilde{T}_{1}, \widetilde{T}_{2}\right]\right)+\left(\widetilde{\delta_{1} T_{2}}-\widetilde{\delta_{2} T_{1}}\right),

[\delta_1 + \widetilde{T}_1, \delta_2 + \widetilde{T}_2] = ([\delta_1, \delta_2] + [\widetilde{T}_1, \widetilde{T}_2]) + (\widetilde{\delta_1T_2} - \widetilde{\delta_2T_1}),

\begin{aligned} {[\delta, \widetilde{T}] X } & =\delta(T \circ X)-T \circ \delta X=\delta T \circ X+T \circ \delta X-T \circ \delta X \\ & =\delta T \circ X=\widetilde{\delta T} X, \quad X \in \mathfrak{J}^{C} . \end{aligned}

\begin{aligned} [\delta, \widetilde{T}]X \!\!\! &=& \!\!\! \delta(T \circ X) - T \circ \delta X = \delta T \circ X + T \circ \delta X - T \circ \delta X \\ \!\!\! &=& \!\!\! \delta T \circ X = \widetilde{\delta T}X, \quad X \in \text{\es {J}}^C. \; \end{aligned}

\lambda(\phi)=-{ }^{t} \phi=-{ }^{t}(\delta+\widetilde{T})=\delta-\widetilde{T} .

\lambda(\phi) = - ^t\phi = - ^t(\delta + \widetilde{T}) = \delta - \widetilde{T}.

=(\delta X, Y)-(\widetilde{T} X, Y)=((\delta-\widetilde{T}) X, Y) \quad X, Y \in \mathfrak{J}^{C} .

\mathfrak{e}_{6}=\left\{\phi \in \operatorname{Hom}_{C}\left(\mathfrak{J}^{C}\right) \mid(\phi X, X, X)=0,\langle\phi X, Y\rangle+\langle X, \phi Y\rangle=0\right\} .

\text{\es {e}}_6 = \{ \phi \in \text\mathrm{{Hom}}_C(\text{\es {J}}^C) \, | \, (\phi X, X, X) = 0, \langle \phi X, Y \rangle + \langle X, \phi Y \rangle = 0 \}.

\phi=\delta+i \widetilde{T}, \quad \delta \in \mathfrak{f}_{4}, T \in \mathfrak{J}_{0} .

\phi = \delta + i\widetilde{T}, \quad \delta \in \text{\es {f}}_4, T \in \text{\es {J}}_0.

\phi=\frac{\phi-\phi^{*}}{2}+i \frac{\phi+\phi^{*}}{2 i}, \quad \frac{\phi-\phi^{*}}{2}, \frac{\phi+\phi^{*}}{2 i} \in \mathfrak{e}_{6} .

\phi = \frac{\phi - \phi^{*}}{2} + i\frac{\phi + \phi^{*}}{2i}, \qquad \frac{\phi - \phi^{*}}{2}, \frac{\phi + \phi^{*}}{2i} \in \text{\es {e}}_6.

\mathfrak{e}_{6}{ }^{C}=\mathfrak{f}_{4}{ }^{C} \oplus \widetilde{\mathfrak{J}}_{0}^{C} .

{\text{\es {e}}_6}^C = {\text{\es {f}}_4}^C \oplus \widetilde{\text{\es {J}}}_0^{\ C}.

\mathfrak{a} \ni\left[\delta_{1}, \delta+\widetilde{T}\right]=\left[\delta_{1}, \delta\right]+\widetilde{\delta_{1} T} \text { (Theorem 3.2.2), }

\text{\es {a}} \ni [\delta_1, \delta + \widetilde{T}] = [\delta_1, \delta] + \widetilde{\delta_1T}\;\text{(Theorem 3.2.2)},

\mathfrak{a} \supset\left[\mathfrak{a}, \mathfrak{e}_{6}{ }^{C}\right] \supset\left[\mathfrak{f}_{4}{ }^{C}, \widetilde{\mathfrak{J}}_{0}{ }^{C}\right]=\mathfrak{f}_{4} \widetilde{ }^{C} \mathfrak{J}_{0}{ }^{C} \text { (Lemma 3.2.2) }=\widetilde{\mathfrak{J}}_{0}^{C} \text { (Proposition 2.4.6.(2)). }

\text{\es {a}} \supset [\text{\es {a}}, {\text{\es {e}}_6}^C] \supset [{\text{\es {f}}_4}^C, {\widetilde{\text{\es {J}}}_0}^{\ C}] = \widetilde{{\text{\es {f}}_4}^C{\text{\es {J}}_0}^C}\; \text{(Lemma 3.2.2)}\; = {\widetilde{\text{\es {J}}}_0}^{\ C} \; \text{(Proposition 2.4.6.(2))}.

A \circ X=\left(A-\frac{1}{3} \operatorname{tr}(A) E\right)^{\sim} X+\frac{1}{3} \operatorname{tr}(A) X \in W .

A \circ X = \Big(A - \frac{1}{3}\text\mathrm{{tr}}(A)E \Big)^{\sim}X + \frac{1}{3}\text\mathrm{{tr}}(A)X \in W.

A \vee B=[\widetilde{A}, \widetilde{B}]+\left(A \circ B-\frac{1}{3}(A, B) E\right)^{\sim}

A \vee B = [\widetilde{A}, \widetilde{B}] + \Big(A \circ B - \frac{1}{3}(A, B)E \Big)^{\sim}

(A \vee B) X=\frac{1}{2}(B, X) A+\frac{1}{6}(A, B) X-2 B \times(A \times X), \quad X \in \mathfrak{J}^{C} .

(A \vee B)X = \frac{1}{2}(B, X)A + \frac{1}{6}(A, B)X - 2B \times (A \times X), \quad X \in \text{\es {J}}^C.

X \circ(X \circ X)-\operatorname{tr}(X) X \circ X+\frac{1}{2}\left(\operatorname{tr}(X)^{2}-(X, X)\right) X=\frac{1}{3}(X, X, X) E .

X \circ (X \circ X) - \text\mathrm{{tr}}(X)X \circ X + \frac{1}{2}(\text\mathrm{{tr}}(X)^2 - (X, X))X = \frac{1}{3}(X, X, X)E.

\begin{aligned} A & \circ(B \circ X)+B \circ(X \circ A)+X \circ(A \circ B)-\operatorname{tr}(A) B \circ X-\operatorname{tr}(B) A \circ X \\ & -\operatorname{tr}(X) A \circ B+\frac{1}{2}(\operatorname{tr}(A) \operatorname{tr}(B) X+\operatorname{tr}(B) \operatorname{tr}(X) A+\operatorname{tr}(X) \operatorname{tr}(A) B) \\ & -\frac{1}{2}((A, B) X+(B, X) A+(X, A) B)=(A, B, X) E . \end{aligned}

\begin{aligned} & \frac{1}{2}(B, X) A+\frac{1}{6}(A, B) X-2 B \times(A \times X)+\frac{1}{3}(A, B) X \\ = & \frac{1}{2}(B, X) A+\frac{1}{2}(A, B) X-2 B \circ(A \times X)+\operatorname{tr}(B) A \times X+\operatorname{tr}(A \times X) B \\ & -(\operatorname{tr}(B) \operatorname{tr}(A \times X)-(B, A \times X)) E \\ = & \frac{1}{2}(B, X) A+\frac{1}{2}(A, B) X-2 B \circ(A \circ X)+\operatorname{tr}(A) B \circ X+\operatorname{tr}(X) B \circ A \\ & -\operatorname{tr}(A) \operatorname{tr}(X) B+(A, X) B+\operatorname{tr}(B) A \circ X-\frac{1}{2} \operatorname{tr}(B) \operatorname{tr}(A) X \\ & -\frac{1}{2} \operatorname{tr}(B) \operatorname{tr}(X) A+\frac{1}{2} \operatorname{tr}(B) \operatorname{tr}(A) \operatorname{tr}(X) E-\frac{1}{2} \operatorname{tr}(B)(A, X) E \\ & +\frac{1}{2}(\operatorname{tr}(A) \operatorname{tr}(X)-(A, X)) B-\frac{1}{2} \operatorname{tr}(B)(\operatorname{tr}(A) \operatorname{tr}(X)-(A, X)) E+(A, B, X) E \\ = & \operatorname{tr}(A) B \circ X+\operatorname{tr}(B) X \circ A+\operatorname{tr}(X) A \circ B \\ & -\frac{1}{2}(\operatorname{tr}(A) \operatorname{tr}(B) X+\operatorname{tr}(B) \operatorname{tr}(X) A+\operatorname{tr}(A) \operatorname{tr}(X) B) \\ & +\frac{1}{2}((A, B) X+(B, X) A+(A, X) B)+(A, B, X) E-2 B \circ(A \circ X) \\ = & A \circ(B \circ X)+B \circ(A \circ X)+X \circ(A \circ B)-2 B \circ(A \circ X) \\ = & A \circ(B \circ X)-B \circ(A \circ X)+(A \circ B) \circ X \\ = & {[\widetilde{A}, \widetilde{B}] X+(A \circ B)^{\sim} X . } \end{aligned}

\[ \left(E_{1} \vee E_{1}\right)\left(\begin{array}{ccc} \xi_{1} & x_{3} & \bar{x}_{2} \\ \bar{x}_{3} & \xi_{2} & x_{1} \\ x_{2} & \bar{x}_{1} & \xi_{3} \end{array}\right)=\frac{1}{6}\left(\begin{array}{ccc} 4 \xi_{1} & x_{3} & \bar{x}_{2} \\ \bar{x}_{3} & -2 \xi_{2} & -2 x_{1} \\ x_{2} & -2 \bar{x}_{1} & -2 \xi_{3} \end{array}\right) . \] \end{itemize}

(E_1 \vee E_1)\pmatrix{\xi_1 & x_3 & \overline{x}_2 \cr \overline{x}_3 & \xi_2 & x_1 \cr x_2 & \overline{x}_1 & \xi_3} = \frac{1}{6}\pmatrix{4\xi_1 & x_3 & \overline{x}_2 \cr \overline{x}_3 & -2\xi_2 & -2x_1 \cr x_2 & -2\overline{x}_1 & -2\xi_3}.

\left(\phi^{\prime} X, Y\right)=-(X, \phi Y), \quad X, Y \in \mathfrak{J}^{C} .

(\phi'X, Y) = -(X, \phi Y), \quad X, Y \in \text{\es {J}}^C.

\phi(X \times Y)=\phi^{\prime} X \times Y+X \times \phi^{\prime} Y, \quad X, Y \in \mathfrak{J}^{C} .

\phi(X \times Y) = \phi'X \times Y + X \times \phi'Y, \quad X, Y \in \text{\es {J}}^C.

\[ (A \vee B)^{\prime}=-B \vee A . \] \end{itemize}

(A \vee B)' = - B \vee A.

[\phi, A \vee B]=\phi A \vee B+A \vee \phi^{\prime} B .

[\phi, A \vee B] = \phi A \vee B + A \vee \phi'B.

\begin{aligned} = & \phi\left(\frac{1}{2}(B, X) A+\frac{1}{6}(A, B) X-2 B \times(A \times X)\right)-(A \vee B) \phi X \text { (Lemma 3.4.1) } \\ = & \frac{1}{2}(B, X) \phi A+\frac{1}{6}(A, B) \phi X-2 \phi^{\prime} B \times(A \times X)-2 B \times(\phi A \times X) \\ & -2 B \times(A \times \phi X)-\frac{1}{2}(B, \phi X) A-\frac{1}{6}(A, B) \phi X+2 B \times(A \times \phi X) \text { (Lemma 3.4.3) } \\ = & \frac{1}{2}(B, X) \phi A+\frac{1}{6}(\phi A, B) X-2 B \times(\phi A \times X) \\ & +\frac{1}{2}\left(\phi^{\prime} B, X\right) A+\frac{1}{6}\left(A, \phi^{\prime} B\right) X-2 \phi^{\prime} B \times(A \times X) \\ = & (\phi A \vee B) X+\left(A \vee \phi^{\prime} B\right) X, \quad X \in \mathfrak{J}^{C} . \end{aligned}

\left(\phi_{1}, \phi_{2}\right)_{6}=\left(\delta_{1}, \delta_{2}\right)_{4}+\left(T_{1}, T_{2}\right),

(\phi_1, \phi_2)_6 = (\delta_1, \delta_2)_4 + (T_1, T_2),

\left(\left[\phi, \phi_{1}\right], \phi_{2}\right)_{6}+\left(\phi_{1},\left[\phi, \phi_{2}\right]\right)_{6}=0, \quad \phi, \phi_{i} \in \mathfrak{e}_{6}^{C} .

([\phi, \phi_1], \phi_2)_6 + (\phi_1, [\phi, \phi_2])_6 = 0, \quad \phi, \phi_i \in {\text{\es {e}}_6}^C.

\[ (\phi, A \vee B)_{6}=(\phi A, B) . \] \end{itemize}

(\phi, A \vee B)_6 = (\phi A, B).

\begin{aligned} & \left(\left[\phi, \phi_{1}\right], \phi_{2}\right)_{6} \\ & =\left(\left[\delta+\widetilde{T}, \delta_{1}+\widetilde{T}_{1}\right], \delta_{2}+\widetilde{T}_{2}\right)_{6} \end{aligned}

\begin{aligned} & =\left(\left(\left[\delta, \delta_{1}\right]+\left[\widetilde{T}, \widetilde{T}_{1}\right]\right)+\left(\widetilde{\delta T_{1}}-\widetilde{\delta_{1} T}\right), \delta_{2}+\widetilde{T}_{2}\right)_{6} \quad(\text { Theorem 3.2.2 }) \\ & =\left(\left[\delta, \delta_{1}\right], \delta_{2}\right)_{4}+\left(\left[\widetilde{T}, \widetilde{T}_{1}\right], \delta_{2}\right)_{4}+\left(\delta T_{1}-\delta_{1} T, T_{2}\right) \\ & =-\left(\delta_{1},\left[\delta, \delta_{2}\right]\right)_{4}+\left(\delta_{2} T, T_{1}\right)+\left(\delta T_{1}, T_{2}\right)-\left(\delta_{1} T, T_{2}\right) \quad(\text { Lemma } 2.5 .2) \\ & =-\left(\delta_{1},\left[\delta, \delta_{2}\right]\right)_{4}-\left(\delta_{1} T, T_{2}\right)-\left(T_{1}, \delta T_{2}\right)+\left(T_{1}, \delta_{2} T\right) \\ & \left.=-\left(\delta_{1}+\widetilde{T}_{1},\left[\delta, \delta_{2}\right]+\left[\widetilde{T}, \widetilde{T}_{2}\right]\right)+\left(\widetilde{\delta T_{2}}-\widetilde{\delta_{2} T}\right)\right)_{6} \\ & =-\left(\delta_{1}+\widetilde{T}_{1},\left[\delta+\widetilde{T}, \delta_{2}+\widetilde{T}_{2}\right]\right)_{6} \\ & =-\left(\phi_{1},\left[\phi, \phi_{2}\right]\right)_{6} \end{aligned}

\begin{aligned} (\phi, A \vee B)_{6} & =\left(\delta+\widetilde{T},[\widetilde{A}, \widetilde{B}]+\left(A \circ B-\frac{1}{3}(A, B) E\right)^{\sim}\right)_{6} \\ & =(\delta,[\widetilde{A}, \widetilde{B}])_{4}+\left(T, A \circ B-\frac{1}{3}(A, B) E\right) \\ & =(\delta A, B)+(\widetilde{T} A, B)=((\delta+\widetilde{T}) A, B)=(\phi A, B) . \end{aligned}

\begin{aligned} (\phi, A \vee B)_6 \!\!\!&=&\!\!\! \Big(\delta + \widetilde{T}, [\widetilde{A},\widetilde{B}] + \Big(A \circ B - \frac{1}{3}(A, B)E \Big)^{\sim} \Big)_6 \vspace{1mm}\\ \!\!\!&=&\!\!\! (\delta, [\widetilde{A}, \widetilde{B}])_4 + \Big(T, A \circ B - \frac{1}{3}(A, B)E\Big) \vspace{1mm}\\ \!\!\!&=&\!\!\! (\delta A, B) + (\widetilde{T}A, B) = ((\delta + \widetilde{T})A, B) = (\phi A, B). \end{aligned}

\begin{array}{ll} {\left[\left(E_{i}-E_{i+1}\right)^{\sim}, D\right]=0, D \in \mathfrak{d}_{4}^{C},} & {\left[\left(E_{i}-E_{i+1}\right)^{\sim},\left(E_{j}-E_{j+1}\right)^{\sim}\right]=0,} \\ {\left[\left(E_{i}-E_{i+1}\right)^{\sim}, \widetilde{A}_{i}(a)\right]=-\frac{1}{2} \widetilde{F}_{i}(a),} & {\left[\left(E_{i}-E_{i+1}\right)^{\sim}, \widetilde{F}_{i}(a)\right]=-\frac{1}{2} \widetilde{A}_{i}(a),} \\ {\left[\left(E_{i}-E_{i+1}\right)^{\sim}, \widetilde{A}_{i+1}(a)\right]=-\frac{1}{2} \widetilde{F}_{i+1}(a),} & {\left[\left(E_{i}-E_{i+1}\right)^{\sim}, \widetilde{F}_{i+1}(a)\right]=-\frac{1}{2} \widetilde{A}_{i+1}(a),} \\ {\left[\left(E_{i}-E_{i+1}\right)^{\sim}, \widetilde{A}_{i+2}(a)\right]=\widetilde{F}_{i+2}(a),} & {\left[\left(E_{i}-E_{i+1}\right)^{\sim}, \widetilde{F}_{i+2}(a)\right]=\widetilde{A}_{i+2}(a) .} \end{array}

\begin{array}{ll} [(E_i - E_{i+1})^{\sim}, D] = 0, \, D \in {\text{\es {d}}_4}^C, \!\!&\!\! [(E_i - E_{i+1})^{\sim}, (E_j - E_{j+1})^{\sim}] = 0, \vspace{1mm}\\ {[}(E_i - E_{i+1})^{\sim}, \widetilde{A}_i(a){]} = - {\frac{1}{2}}\widetilde{F}_i(a), \!\!&\!\! [(E_i - E_{i+1})^{\sim}, \widetilde{F}_i(a)] = - {\frac{1}{2}}\widetilde{A}_i(a), \vspace{1mm}\\ {[}(E_i - E_{i+1})^{\sim}, \widetilde{A}_{i+1}(a){]} = - {\frac{1}{2}}\widetilde{F}_{i+1}(a), \!\!&\!\! [(E_i - E_{i+1})^{\sim}, \widetilde{F}_{i+1}(a)] = - {\frac{1}{2}}\widetilde{A}_{i+1}(a), \vspace{1mm}\\ {[}(E_i - E_{i+1})^{\sim}, \widetilde{A}_{i+2}(a){]} = \widetilde{F}_{i+2}(a), \!\!&\!\! [(E_i - E_{i+1})^{\sim}, \widetilde{F}_{i+2}(a)] = \widetilde{A}_{i+2}(a). \end{array} \vspace{2mm}

\begin{aligned} B_{6}\left(\phi_{1}, \phi_{2}\right) & =12\left(\phi_{1}, \phi_{2}\right)_{6} \\ & =12\left(\delta_{1}, \delta_{2}\right)_{4}+12\left(T_{1}, T_{2}\right) \\ & =\frac{4}{3} B_{4}\left(\delta_{1}, \delta_{2}\right)+12\left(T_{1}, T_{2}\right) \\ & =4 \operatorname{tr}\left(\phi_{1} \phi_{2}\right) \end{aligned}

\begin{aligned} B_6(\phi_1, \phi_2) \!\!\! &=& \!\!\! 12(\phi_1, \phi_2)_6 \vspace{1mm}\\ \!\!\! &=& \!\!\! 12(\delta_1, \delta_2)_4 + 12(T_1, T_2) \vspace{1mm}\\ \!\!\! &=& \!\!\! \frac{4}{3}B_4(\delta_1, \delta_2) + 12(T_1, T_2) \vspace{1mm}\\ \!\!\! &=& \!\!\! 4\text\mathrm{{tr}}(\phi_1 \phi_2), \end{aligned}

B_{6}\left(\phi_{1}, \phi_{2}\right)=k\left(\phi_{1}, \phi_{2}\right)_{6}=k^{\prime} \operatorname{tr}\left(\phi_{1} \phi_{2}\right) .

B_6(\phi_1, \phi_2) = k(\phi_1, \phi_2)_6 = k'\text\mathrm{{tr}}(\phi_1\phi_2).

(\phi, \phi)_{6}=\left(\left(E_{1}-E_{2}\right)^{\sim},\left(E_{1}-E_{2}\right)^{\sim}\right)_{6}=\left(E_{1}-E_{2}, E_{1}-E_{2}\right)=2 .

(\phi, \phi)_6 = ((E_1 - E_2)^{\sim}, (E_1 - E_2)^{\sim})_6 = (E_1 - E_2, E_1 - E_2) = 2.

\left[\phi,\left[\phi, \widetilde{A}_{1}\left(e_{i}\right)\right]\right]=\left[\phi,-\frac{1}{2} \widetilde{F}_{1}\left(e_{i}\right)\right]=\frac{1}{4} \widetilde{A}_{1}\left(e_{i}\right),

\begin{aligned} & {\left[\phi,\left[\phi, \widetilde{A}_{2}\left(e_{i}\right)\right]\right]=\left[\phi,-\frac{1}{2} \widetilde{F}_{2}\left(e_{i}\right)\right]=\frac{1}{4} \widetilde{A}_{2}\left(e_{i}\right),} \\ & {\left[\phi,\left[\phi, \widetilde{A}_{3}\left(e_{i}\right)\right]\right]=\left[\phi, \widetilde{F}_{3}\left(e_{i}\right)\right]=\widetilde{A}_{3}\left(e_{i}\right),} \\ & {\left[\phi,\left[\phi, \widetilde{F}_{1}\left(e_{i}\right)\right]\right]=\left[\phi,-\frac{1}{2} \widetilde{A}_{1}\left(e_{i}\right)\right]=\frac{1}{4} \widetilde{F}_{1}\left(e_{i}\right),} \\ & {\left[\phi,\left[\phi, \widetilde{F}_{2}\left(e_{i}\right)\right]\right]=\left[\phi,-\frac{1}{2} \widetilde{A}_{2}\left(e_{i}\right)\right]=\frac{1}{4} \widetilde{F}_{2}\left(e_{i}\right),} \\ & {\left[\phi,\left[\phi, \widetilde{F}_{3}\left(e_{i}\right)\right]\right]=\left[\phi, \widetilde{A}_{3}\left(e_{i}\right)\right]=\widetilde{F}_{3}\left(e_{i}\right),} \end{aligned}

B_{6}(\phi, \phi)=\operatorname{tr}\left((\operatorname{ad} \phi)^{2}\right)=\left(\frac{1}{4} \times 4+1 \times 2\right) \times 8=24 .

B_6(\phi, \phi) = \text\mathrm{{tr}}((\text{ad}\phi)^2) = \Big({\frac{1}{4}} \times 4 + 1 \times 2 \Big) \times 8 = 24.

\begin{array}{ll} \phi \phi E_{1}=\phi E_{1}=E_{1}, & \phi \phi \widetilde{F}_{1}\left(e_{i}\right)=-\frac{1}{2} \phi \widetilde{F}_{1}\left(e_{i}\right)=\frac{1}{4} \widetilde{F}_{1}\left(e_{i}\right), \\ \phi \phi E_{2}=-\phi E_{2}=E_{2}, & \phi \phi \widetilde{F}_{2}\left(e_{i}\right)=\frac{1}{2} \phi \widetilde{F}_{2}\left(e_{i}\right)=\frac{1}{4} \widetilde{F}_{2}\left(e_{i}\right), \\ \phi \phi E_{3}=\phi 0=0, & \phi \phi \widetilde{F}_{3}\left(e_{i}\right)=\phi 0=0 . \end{array}

\begin{array}{ll} \phi\phi E_1 = \phi E_1 = E_1, & \phi\phi\widetilde{F}_1(e_i) = - {\frac{1}{2}}\phi\widetilde{F}_1(e_i) = {\frac{1}{4}}\widetilde{F}_1(e_i), \vspace{1mm}\\ \phi\phi E_2 = - \phi E_2 = E_2, & \phi\phi\widetilde{F}_2(e_i) = {\frac{1}{2}}\phi\widetilde{F}_2(e_i) = {\frac{1}{4}}\widetilde{F}_2(e_i), \vspace{1mm}\\ \phi\phi E_3 = \phi 0 = 0, & \phi\phi\widetilde{F}_3(e_i) = \phi 0 = 0. \end{array}

\operatorname{tr}(\phi \phi)=1 \times 2+\frac{1}{4} \times 8 \times 2=6 .

\text\mathrm{{tr}}(\phi\phi) = 1 \times 2 + \frac{1}{4} \times 8 \times 2 = 6.

A \vee(B \times C)+B \vee(C \times A)+C \vee(A \times B)=0, \quad A, B, C \in \mathfrak{J}^{C} .

\begin{aligned} & =2\left(\phi^{\prime} A \times A, A\right)(\text { Lemma } 3.4 .3 .(1))=2\left(\phi^{\prime} A, A \times A\right)=-2(A, \phi(A \times A)) \\ & =-2(\phi,(A \times A) \vee A)_{6} . \end{aligned}

\mathfrak{M}^{r}=\{X \in M(3, \mathfrak{C}) \mid \text { all diagonal elements of } X \text { are real }\},

\text{\es {M}}^r = \{ X \in M(3,\text{\es {C}}) \, | \, \text{all diagonal elements of }\, X \, \text{are real}\},

X \circ Y=\frac{1}{2}\left(X Y+Y^{*} X^{*}\right)

X \circ Y = \frac{1}{2}(XY + Y^*X^*),

\delta\left(\begin{array}{ccc} \xi_{1} & x_{12} & x_{13} \\ x_{21} & \xi_{2} & x_{23} \\ x_{31} & x_{32} & \xi_{3} \end{array}\right)=\left(\begin{array}{ccc} 0 & \delta_{3} x_{12} & \bar{\delta}_{2} \bar{x}_{13} \\ \bar{\delta}_{3} \bar{x}_{21} & 0 & \delta_{1} x_{23} \\ \delta_{2} x_{31} & \bar{\delta}_{1} \bar{x}_{32} & 0 \end{array}\right) .

\delta\pmatrix{\xi_1 & x_{12} & x_{13} \cr x_{21} & \xi_2 & x_{23} \cr x_{31} & x_{32} & \xi_3} = \pmatrix{0 & \delta_3x_{12} & \overline{\delta_2\overline{x}_{13}} \vspace{0.5mm}\cr \overline{\delta_3\overline{x}_{21}} & 0 & \delta_1x_{23} \vspace{0.5mm}\cr \delta_2 x_{31} & \overline{\delta_1\overline{x}_{32}} & 0}.

\delta(X \circ Y)=\delta X \circ Y+X \circ \delta Y, \quad X, Y \in\left(\mathfrak{M}^{r}\right)^{C} .

\delta(X \circ Y) = \delta X \circ Y + X \circ \delta Y, \quad X, Y \in (\text{\es {M}}^r)^C.

\begin{array}{lll} \sigma_{23}=\delta_{1}, & \sigma_{31}=\delta_{2}, & \sigma_{12}=\delta_{3} \\ \sigma_{32}=\kappa \delta_{1}, & \sigma_{13}=\kappa \delta_{2}, & \sigma_{21}=\kappa \delta_{3} . \end{array}

\begin{aligned} & R\left(\sum_{k=1}^{3}\left(\left(\sigma_{i k} x_{i k}\right) y_{k i}+\bar{y}_{k i}\left(\overline{\sigma_{i k} x_{i k}}\right)\right)+\sum_{k=1}^{3}\left(x_{i k}\left(\sigma_{k i} y_{k i}\right)+\left(\overline{\sigma_{k i} y_{k i}}\right) \bar{x}_{i k}\right)\right) \\ & \quad=2 R\left(\sum_{k}\left(\left(\sigma_{i k} x_{i k}\right) y_{k i}+x_{i k}\left(\sigma_{k i} y_{k i}\right)\right)\right) \\ & \quad=2 \sum_{k}\left(\left(\sigma_{i k} x_{i k}, \bar{y}_{k i}\right)+\left(x_{i k}, \overline{\sigma_{k i} y_{k i}}\right)\right) \\ & \quad=2 \sum_{k}\left(\left(x_{i k},-\sigma_{i k} \bar{y}_{k i}\right)+\left(x_{i k}, \sigma_{i k} \bar{y}_{k i}\right)\right)=0 . \end{aligned}

\begin{aligned} & \sum_{k=1}^{3}\left(\left(\sigma_{i k} x_{i k}\right) y_{k j}+\bar{y}_{k i}\left(\overline{\sigma_{j k} x_{j k}}\right)\right)+\sum_{k=1}^{3}\left(x_{i k}\left(\sigma_{k j} y_{k j}\right)+\left(\overline{\sigma_{k i} y_{k i}}\right) \bar{x}_{j k}\right) \\ & =\sum_{k}\left(\left(\sigma_{i k} x_{i k}\right) y_{k j}+y_{i k}\left(\sigma_{k j} x_{k j}\right)\right)+\sum_{k} \overline{\left(\left(\sigma_{j k} x_{j k}\right) y_{k i}+x_{j k}\left(\sigma_{k i} y_{k i}\right)\right)} \end{aligned}

\left(\sigma_{i k} x_{i k}\right) y_{k j}+x_{i k}\left(\sigma_{k j} y_{k j}\right)=\sigma_{i j}\left(x_{i k} y_{k j}\right) .

(\sigma_{ik}x_{ik})y_{kj} + x_{ik}(\sigma_{kj}y_{kj}) = \sigma_{ij}(x_{ik}y_{kj}).

\begin{aligned} & =\sum_{k} \sigma_{i j}\left(x_{i k} y_{k j}\right)+\sum_{k} \overline{\sigma_{j i}\left(x_{j k} y_{k i}\right)} \\ & =\sum_{k} \sigma_{i j}\left(x_{i k} y_{k j}\right)+\sum_{k} \sigma_{i j}\left(\overline{x_{j k} y_{k i}}\right)=(i, j) \text {-element of } \delta(X \circ Y) . \end{aligned}

\begin{array}{l} = \sum_{k}\sigma_{ij}(x_{ik}y_{kj}) + \sum_{k}\overline{\sigma_{ji}(x_{jk}y_{ki})} \qquad \vspace{1.5mm}\\ = \sum_{k}\sigma_{ij}(x_{ik}y_{kj}) + \sum_{k}\sigma_{ij}(\overline{x_{jk}y_{ki}}) = (i,j)\text{-element of}\; \, \delta(X \circ Y).\qquad \end{array}

\widetilde{T} X=\frac{1}{2}\left(T X+X T^{*}\right), \quad \text { where } \quad T^{*}={ }^{t} \bar{T} .

\widetilde{T}X = \frac{1}{2}(TX + XT^*), \quad \text{where} \quad T^* = {}^t\overline{T}.

[\delta, \widetilde{R}]=\widetilde{\delta R} .

[\delta, \widetilde{R} ]=\widetilde{\delta R}.

\[ [\widetilde{H}, \widetilde{T}]=\frac{1}{2}[H, T]^{\sim} . \] \end{itemize}

[\widetilde{H}, \widetilde{T}] = \frac{1}{2}[H, T]^{\sim}.

\[ \begin{aligned} & =\frac{1}{2}\left(\widetilde{H}\left(T X+X T^{*}\right)-\widetilde{T}\left(H X+X H^{*}\right)\right) \\ & =\frac{1}{4}\left(H T X+X T^{*} H^{*}+H X T^{*}+T X H^{*}-T H X-X H^{*} T^{*}-T X H^{*}-H X T^{*}\right) \end{aligned} \] \end{itemize}

=\frac{1}{4}\left([H, T] X+X[H, T]^{*}\right)=\frac{1}{2}[H, T]^{\sim} X, \quad X \in \mathfrak{J}^{C} .

\begin{aligned} & \pm\left(\lambda_{k}-\lambda_{l}\right), \quad \pm\left(\lambda_{k}+\lambda_{l}\right), \quad 0 \leq k<l \leq 3, \\ & \quad \pm \lambda_{k} \pm \frac{1}{2}\left(\mu_{2}-\mu_{3}\right), \quad 0 \leq k \leq 3, \\ & \pm \frac{1}{2}\left(-\lambda_{0}-\lambda_{1}+\lambda_{2}-\lambda_{3}\right) \pm \frac{1}{2}\left(\mu_{3}-\mu_{1}\right), \\ & \pm \frac{1}{2}\left(\lambda_{0}+\lambda_{1}+\lambda_{2}-\lambda_{3}\right) \pm \frac{1}{2}\left(\mu_{3}-\mu_{1}\right), \\ & \pm \frac{1}{2}\left(-\lambda_{0}+\lambda_{1}+\lambda_{2}+\lambda_{3}\right) \pm \frac{1}{2}\left(\mu_{3}-\mu_{1}\right), \\ & \pm \frac{1}{2}\left(\lambda_{0}-\lambda_{1}+\lambda_{2}+\lambda_{3}\right) \pm \frac{1}{2}\left(\mu_{3}-\mu_{1}\right), \\ & \pm \frac{1}{2}\left(\lambda_{0}-\lambda_{1}+\lambda_{2}-\lambda_{3}\right) \pm \frac{1}{2}\left(\mu_{1}-\mu_{2}\right), \\ & \pm \frac{1}{2}\left(-\lambda_{0}+\lambda_{1}+\lambda_{2}-\lambda_{3}\right) \pm \frac{1}{2}\left(\mu_{1}-\mu_{2}\right), \\ & \pm \frac{1}{2}\left(\lambda_{0}+\lambda_{1}+\lambda_{2}+\lambda_{3}\right) \pm \frac{1}{2}\left(\mu_{1}-\mu_{2}\right), \\ & \pm \frac{1}{2}\left(-\lambda_{0}-\lambda_{1}+\lambda_{2}+\lambda_{3}\right) \pm \frac{1}{2}\left(\mu_{1}-\mu_{2}\right), \end{aligned}

\mathfrak{e}_{6}{ }^{C}=\mathfrak{f}_{4}{ }^{C} \oplus \tilde{\mathfrak{J}}_{0}^{C} .

{\text{\es {e}}_6}^C = {\text{\es {f}}_4}^C \oplus {\widetilde{\text{\es {J}}}_0}^{\ C}.

\mathfrak{h}=\left\{\begin{array}{l|l} h=h_{\delta}+\widetilde{H} \in \mathfrak{e}_{6}^{C} & \begin{array}{l} h_{\delta}=\sum_{k=0}^{3} \lambda_{k} H_{k}=-\sum_{k=0}^{3} \lambda_{k} i G_{k 4+k}, \lambda_{k} \in C \\ H=\sum_{j=1}^{3} \mu_{j} E_{j}, \mu_{j} \in C, \mu_{1}+\mu_{2}+\mu_{3}=0 \end{array} \end{array}\right\},

\begin{aligned} & {\left[h_{\delta}, h_{\delta^{\prime}}\right]=0,} \\ & {\left[h_{\delta}, \widetilde{H}^{\prime}\right]=\widetilde{h_{\delta} H^{\prime}}(\text { Lemma 3.6.3.(1) })=\widetilde{0}=0,} \\ & {\left[\widetilde{H}, \widetilde{H}^{\prime}\right]=\frac{1}{2}\left[H, H^{\prime}\right]^{\sim}(\text { Lemma 3.6.3.(2) })=0 .} \end{aligned}

\begin{aligned} {[h, S] } & =\left[h_{\delta}+\widetilde{H}, S\right]=\left[h_{\delta}, S\right]-[S, \widetilde{H}] \\ & =\alpha\left(h_{\delta}\right) S-\widetilde{S H}(\text { Lemma 3.6.3.(1)) } \\ & =\alpha\left(h_{\delta}\right) S=\left( \pm \lambda_{k} \pm \lambda_{l}\right) S \end{aligned}

\begin{aligned} [h ,S] \!\!\! &=& \!\!\! [h_\delta + \widetilde{H}, S] = [h_\delta, S] - [S, \widetilde{H}] \vspace{1mm}\\ \!\!\! &=& \!\!\! \alpha(h_\delta)S - \widetilde{SH}\;\; \text{(Lemma 3.6.3.(1))}\vspace{1mm}\\ \!\!\! &=& \!\!\! \alpha(h_\delta)S = (\pm \lambda_k \pm \lambda_l)S. \end{aligned}

\begin{aligned} {\left[h, \widetilde{F}_{23}(a)\right] } & =\left[h_{\delta}, \widetilde{F}_{23}(a)\right]+\left[\widetilde{H}, \widetilde{F}_{23}(a)\right] \\ & =\left(h_{\delta} F_{23}(a)\right)^{\sim}+\frac{1}{2}\left[H, F_{23}(a)\right]^{\sim}(\text { Lemma 3.6.3 }) \\ & =\widetilde{F}_{23}\left(h_{\delta} a\right)+\frac{1}{2}\left[H, F_{23}(a)\right]^{\sim} \\ & =\left(\lambda_{k}+\frac{1}{2}\left(\mu_{2}-\mu_{3}\right)\right) \widetilde{F}_{23}(a), \end{aligned}

\begin{aligned} [h, \widetilde{F}_{23}(a)] \!\!\! &=& \!\!\! [h_\delta, \widetilde{F}_{23}(a)] + [\widetilde{H}, \widetilde{F}_{23}(a)] \\ \!\!\! &=& \!\!\! (h_{\delta}F_{23}(a))^{\sim} + {\frac{1}{2}}[H, F_{23}(a)]^{\sim} \;\;\text{(Lemma 3.6.3)} \\ \!\!\! &=& \!\!\! \widetilde{F}_{23}(h_\delta a) + {\frac{1}{2}}[H, F_{23}(a)]^{\sim} \\ \!\!\! &=& \!\!\! \Big(\lambda_k + {\frac{1}{2}}(\mu_2 - \mu_3)\Big)\widetilde{F}_{23}(a), \end{aligned}

\left[h, \widetilde{F}_{32}(a)\right]=\left(-\lambda_{k}+\frac{1}{2}\left(\mu_{3}-\mu_{2}\right)\right) \widetilde{F}_{32}(a), \quad a=e_{k}+i e_{4+k},

[h, \widetilde{F}_{32}(a)] = \Big(- \lambda_k + {\frac{1}{2}}(\mu_3 - \mu_2)\Big)\widetilde{F}_{32}(a), \quad a = e_k + i e_{4+k},

\begin{aligned} & {\left[h, \widetilde{F}_{31}(a)\right]=\left(h_{\delta} F_{31}(a)\right)^{\sim}+\frac{1}{2}\left[\widetilde{H}, \widetilde{F}_{31}(a)\right]=\widetilde{F}_{31}\left(\left(\nu h_{\delta}\right) a\right)+\frac{1}{2}\left(\mu_{3}-\mu_{1}\right) \widetilde{F}_{31}(a),} \\ & {\left[h, \widetilde{F}_{12}(a)\right]=\left(h_{\delta} F_{12}(a)\right)^{\sim}+\frac{1}{2}\left[\widetilde{H}, \widetilde{F}_{12}(a)\right]=\widetilde{F}_{12}\left(\left(\kappa \pi h_{\delta}\right) a\right)+\frac{1}{2}\left(\mu_{1}-\mu_{2}\right) \widetilde{F}_{12}(a),} \end{aligned}

\begin{array}{l} [h, \widetilde{F}_{31}(a)] = (h_\delta F_{31}(a))^{\sim} + {\frac{1}{2}}[\widetilde{H}, \widetilde{F}_{31}(a)] = \widetilde{F}_{31}((\nu h_{\delta})a) + {\frac{1}{2}}(\mu_3 - \mu_1)\widetilde{F}_{31}(a), \vspace{1mm}\\ {[} h, \widetilde{F}_{12}(a){]} = (h_\delta F_{12}(a))^{\sim} + {\frac{1}{2}}[\widetilde{H}, \widetilde{F}_{12}(a)] = \widetilde{F}_{12}((\kappa\pi h_{\delta})a) + {\frac{1}{2}}(\mu_1 - \mu_2)\widetilde{F}_{12}(a), \end{array}

\begin{aligned} \nu h_{\delta} & =\frac{1}{2}\left(-\lambda_{0}-\lambda_{1}+\lambda_{2}-\lambda_{3}\right) H_{0}+\frac{1}{2}\left(\lambda_{0}+\lambda_{1}+\lambda_{2}-\lambda_{3}\right) H_{1} \\ & +\frac{1}{2}\left(-\lambda_{0}+\lambda_{1}+\lambda_{2}+\lambda_{3}\right) H_{2}+\frac{1}{2}\left(\lambda_{0}-\lambda_{1}+\lambda_{2}+\lambda_{3}\right) H_{3} \\ \kappa \pi h_{\delta} & =\frac{1}{2}\left(-\lambda_{0}+\lambda_{1}-\lambda_{2}+\lambda_{3}\right) H_{0}+\frac{1}{2}\left(-\lambda_{0}+\lambda_{1}+\lambda_{2}-\lambda_{3}\right) H_{1} \\ & +\frac{1}{2}\left(\lambda_{0}+\lambda_{1}+\lambda_{2}+\lambda_{3}\right) H_{2}+\frac{1}{2}\left(-\lambda_{0}-\lambda_{1}+\lambda_{2}+\lambda_{3}\right) H_{3} \end{aligned}

\begin{aligned} \nu h_\delta \!\!\! &=& \!\!\! {\frac{1}{2}}(- \lambda_0 - \lambda_1 + \lambda_2 - \lambda_3)H_0 + {\frac{1}{2}}(\lambda_0 + \lambda_1 + \lambda_2 - \lambda_3)H_1 \\ \!\!\! &+& \!\!\! {\frac{1}{2}}(-\lambda_0 + \lambda_1 + \lambda_2 + \lambda_3)H_2 + {\frac{1}{2}}(\lambda_0 - \lambda_1 + \lambda_2 + \lambda_3)H_3, \vspace{1mm}\\ \kappa\pi h_\delta \!\!\! &=& \!\!\! {\frac{1}{2}}(- \lambda_0 + \lambda_1 - \lambda_2 + \lambda_3)H_0 + {\frac{1}{2}}(- \lambda_0 + \lambda_1 + \lambda_2 - \lambda_3)H_1\\ \!\!\! &+& \!\!\! {\frac{1}{2}}(\lambda_0 + \lambda_1 + \lambda_2 + \lambda_3)H_2 + {\frac{1}{2}}(-\lambda_0 - \lambda_1 + \lambda_2 + \lambda_3)H_3 \end{aligned}

\begin{aligned} & \alpha_{1}=\lambda_{0}-\lambda_{1}, \quad \alpha_{2}=\lambda_{1}-\lambda_{2}, \quad \alpha_{3}=\lambda_{2}-\lambda_{3}, \\ & \alpha_{4}=\lambda_{3}+\frac{1}{2}\left(\mu_{2}-\mu_{3}\right), \\ & \alpha_{5}=\frac{1}{2}\left(-\lambda_{0}-\lambda_{1}-\lambda_{2}+\lambda_{3}\right)+\frac{1}{2}\left(\mu_{3}-\mu_{1}\right), \\ & \alpha_{6}=\frac{1}{2}\left(\lambda_{0}+\lambda_{1}+\lambda_{2}+\lambda_{3}\right)+\frac{1}{2}\left(\mu_{1}-\mu_{2}\right) \end{aligned}

\begin{array}{l} \alpha_1 = \lambda_0 - \lambda_1, \quad \alpha_2 = \lambda_1 - \lambda_2, \quad \alpha_3 = \lambda_2 - \lambda_3, \vspace{1mm}\\ \alpha_4 = \lambda_3 + {\frac{1}{2}}(\mu_2 - \mu_3), \vspace{1mm}\\ \alpha_5 = {\frac{1}{2}}(- \lambda_0 - \lambda_1 - \lambda_2 + \lambda_3) + {\frac{1}{2}}(\mu_3 - \mu_1), \vspace{1mm}\\ \alpha_6 = {\frac{1}{2}}(\lambda_0 + \lambda_1 + \lambda_2 + \lambda_3) + {\frac{1}{2}}(\mu_1 - \mu_2) \end{array}

\mu=\alpha_{1}+2 \alpha_{2}+3 \alpha_{3}+2 \alpha_{4}+2 \alpha_{5}+\alpha_{6}

\mu = \alpha_1 + 2\alpha_2 + 3\alpha_3 + 2\alpha_4 + 2\alpha_5 + \alpha_6

\begin{aligned} & \lambda_{0}-\lambda_{1}=1 \quad 0 \quad 0 \quad 0 \quad 0 \quad 0 \\ & \lambda_{0}-\lambda_{2}=1 \quad 1 \quad 0 \quad 0 \quad 0 \quad \lambda_{0}+\lambda_{1}=1 \quad 2 \\ & \lambda_{0}-\lambda_{3}=1 \quad 1 \quad 1 \quad 0 \quad 2 \end{aligned} \quad \frac{1}{1} \text { = } \begin{array}{llllllll} 1 & 1 & 1 & 0 & 0 & \lambda_{0}+\lambda_{3}=1 & 1 & 2 \\ \lambda_{1}-\lambda_{2} & =0 & 1 & 0 & 0 & 0 & 0 & 1 \end{array}

\begin{aligned} & \lambda_{0}+\frac{1}{2}\left(\mu_{2}-\mu_{3}\right)=1 \quad 1 \quad 1 \quad 1 \quad a \quad 0 \\ & \lambda_{1}+\frac{1}{2}\left(\mu_{2}-\mu_{3}\right)=0 \quad 1 \quad 1 \quad 1 \quad a \\ & \lambda_{2}+\frac{1}{2}\left(\mu_{2}-\mu_{3}\right)=0 \quad 0 \quad 1 \quad 1 \\ & \lambda_{3}+\frac{1}{2}\left(\mu_{2}-\mu_{3}\right)=0 \quad 0 \quad 0 \\ & \lambda_{0}-\frac{1}{2}\left(\mu_{2}-\mu_{3}\right)=1 \quad 1 \quad 1 \quad 1 \quad 0 \\ & \lambda_{1}-\frac{1}{2}\left(\mu_{2}-\mu_{3}\right)=0 \quad 1 \quad 1 \quad 1 \quad 0 \\ & \lambda_{2}-\frac{1}{2}\left(\mu_{2}-\mu_{3}\right)=0 \quad 0 \quad 1 \quad 1 \\ & \lambda_{3}-\frac{1}{2}\left(\mu_{2}-\mu_{3}\right)=0 \quad 0 \quad 0 \quad 0 \end{aligned}

\begin{aligned} & \frac{1}{2}\left(-\lambda_{0}-\lambda_{1}+\lambda_{2}-\lambda_{3}\right)+\frac{1}{2}\left(\mu_{3}-\mu_{1}\right)=0 \quad 0 \quad 1 \quad 0 \quad 1 \\ & \frac{1}{2}\left(\lambda_{0}+\lambda_{1}+\lambda_{2}-\lambda_{3}\right)+\frac{1}{2}\left(\mu_{3}-\mu_{1}\right)=1 \quad 2 \quad 3 \quad 1 \quad 1 \quad 2 \\ & \frac{1}{2}\left(-\lambda_{0}+\lambda_{1}+\lambda_{2}+\lambda_{3}\right)+\frac{1}{2}\left(\mu_{3}-\mu_{1}\right)=0 \quad 1 \quad 2 \quad 1 \\ & \frac{1}{2}\left(\quad \lambda_{0}-\lambda_{1}+\lambda_{2}+\lambda_{3}\right)+\frac{1}{2}\left(\mu_{3}-\mu_{1}\right)=1 \quad 1 \quad 2 \quad 2 \quad 1 \quad 2 \\ & \frac{1}{2}\left(\quad \lambda_{0}+\lambda_{1}-\lambda_{2}+\lambda_{3}\right)+\frac{1}{2}\left(\mu_{3}-\mu_{1}\right)=1 \quad 2 \quad 2 \quad 1 \\ & \frac{1}{2}\left(-\lambda_{0}-\lambda_{1}-\lambda_{2}+\lambda_{3}\right)+\frac{1}{2}\left(\mu_{3}-\mu_{1}\right)=0 \quad 0,0 \quad 0 \\ & \frac{1}{2}\left(\quad \lambda_{0}-\lambda_{1}-\lambda_{2}-\lambda_{3}\right)+\frac{1}{2}\left(\mu_{3}-\mu_{1}\right)=1 \quad 1 \quad 1 \quad 1 \quad 0 \quad 1 \\ & \frac{1}{2}\left(-\lambda_{0}+\lambda_{1}-\lambda_{2}-\lambda_{3}\right)+\frac{1}{2}\left(\mu_{3}-\mu_{1}\right)=0 \quad 1 \quad 1 \quad 0 \\ & \frac{1}{2}\left(\quad \lambda_{0}-\lambda_{1}+\lambda_{2}-\lambda_{3}\right)-\frac{1}{2}\left(\mu_{1}-\mu_{2}\right)=1 \quad 1 \quad 2 \quad 1 \\ & \frac{1}{2}\left(\quad \lambda_{0}-\lambda_{1}-\lambda_{2}+\lambda_{3}\right)-\frac{1}{2}\left(\mu_{1}-\mu_{2}\right)=1 \quad 1 \quad 1 \end{aligned} \quad \frac{1}{1} \quad 1 \quad 0