LaTeX vs MathPix image — killing1889

\left(X_{i} X_{x}\right)=\sum ? c_{t \times p} X_{p} f,

\sum \ell\left\{c_{x 2_{p}}\left(X_{p} X_{i}\right)+c_{2_{i \ell}}\left(X_{p} X_{x}\right)+c_{i x \rho}\left(X_{\rho} X_{2}\right)\right\}=0,

\left(\sum \eta_{i} X_{i}, \sum \xi_{i} X_{i}\right)=\sum_{i \times \rho} \eta_{i} \xi_{x} c_{i \times \rho} X_{\rho} f=\omega \sum \xi_{\rho} X_{\rho} f .

\frac{\left.\left\lvert\, \begin{array}{ccc} \sum \eta_{i} c_{i 11}-\omega & \sum \eta_{i} c_{i 21} & \cdots \end{array}\right.\right] \eta_{i} c_{i r 1}}{\sum \eta_{i} c_{i 12}}\left[\left.\begin{array}{ccc} \sum \eta_{i} c_{i 22}-\omega & \cdots & \sum \eta_{i} c_{i r 2} \\ \sum \eta_{i} c_{i 1 r} & \sum \eta_{i} c_{i 2 r} & \cdots \sum \eta_{i} c_{i r r}-\omega \end{array} \right\rvert\,\right.

\omega^{r}-\omega^{r-1} \psi_{1}(\eta)+\omega^{r-2} \psi_{2}(\eta)+\cdots \pm \omega \psi_{r-1}(\eta)=0

\left(X_{r} X_{r-1}\right)=\varepsilon X_{r} f,

\left(X_{r} X_{1}\right)=\omega X_{1} f

\sum_{1}^{m} e\left(c_{r-1, x \rho} c_{r \rho x}-c_{r-1, \rho x} c_{r x \rho}\right)=\omega \varepsilon .

0=m \omega \varepsilon,

\left(\sum \eta_{i} X_{i}, \quad \sum \xi_{i} X_{i}\right)=\varepsilon \sum \eta_{i} X_{i} f .

\left(X_{r} X_{r-1}\right)=0, \cdots,\left(X_{r} X_{r-1+1}\right)=0 .

\begin{aligned} & \left(X_{r} X_{r-i}\right)=\left[X_{r} X_{r-1} \cdots X_{r-1+1}\right], \\ & \left(X_{r} X_{r-i-1}\right)=\left[X_{r} X_{r-1} \cdots X_{r-i+1} X_{r-1}\right], \\ & \left(X_{r} X_{r-k+1}\right)=\left[X_{r} X_{r-1} \cdots X_{r-k+2}\right] . \end{aligned}

\left(X_{r} X_{\alpha}\right)=\sum e_{\alpha \beta} X_{\beta},

\left(X_{r}\left(X_{r-x} X_{r-x}\right)\right)=0,

\left(X_{r-x} X_{r-2}\right)=\left[X_{r} X_{r-1} \ldots X_{r-i+1}\right] .

\left(X_{r}\left(X_{r-2} X_{\alpha}\right)\right)=\sum e_{\alpha \beta}\left(X_{r-2} X_{\beta}\right) .

\left(X_{r-2} X_{\alpha}\right)=\left[X_{1} \ldots X_{r-k}\right],

\left(X_{r},\left(X_{r-2} X_{r-i}\right)\right)=\left(X_{r-2}\left[X_{r} \ldots X_{r-i+1}\right]\right),

\left(X_{r},\left(X_{r-2} X_{r-1}\right)\right)=\left[X_{r} \ldots X_{r-i+1}\right]

\left(X_{r-2} X_{r-i}\right)=\left[X_{r} X_{r-1} \ldots X_{r-i}\right] .

\left\{\begin{array}{l} \left(X_{r-2} X_{r-i-1}\right)=\left[X_{r} X_{r-1} \ldots X_{r-i-1}\right], \\ \left.\left(X_{r-2} X_{r-k+1}\right)=\left[X_{r} X_{r-1} \ldots X_{r-k+1}\right] .^{*}\right) \end{array}\right.

\begin{gathered} \left(X_{r}\left(X_{r-i-v} X_{r-i-p}\right)\right) \\ =\left(X_{r-i-r}\left[X_{r} \ldots X_{r-i-p+i}\right]\right)-\left(X_{r-i-p}\left[X_{r} \ldots X_{r-i-r+1}\right]\right), \end{gathered}

\left(X_{r-i-v} X_{r-i-p}\right)=\left[X_{r} \ldots X_{r-k+1}\right] .

\left(\mu_{0} X_{r}+\mu_{1} X_{r-1}+\cdots+\mu_{k-1} X_{r-k+1}, \nu_{1} X_{1}+\nu_{2} X_{2}+\cdots+\nu_{r-k} X_{r-k}\right)=0

\eta_{1}=\eta_{2}=\cdots=\eta_{r-k}=0, \quad \eta_{r-k+1}=\mu_{k-1} \cdots \eta_{r}=\mu_{0}

\begin{gathered} \left(X_{r} X_{r-1}\right)=a_{1} X_{r-2} f, \quad\left(X_{r} X_{r-2}\right)=a_{2} X_{r-3} f \ldots \\ \left(X_{r} X_{r-k+2}\right)=a_{k-2} X_{r-k+1}, \quad\left(X_{r} X_{r-k+1}\right)=0 . \end{gathered}

\left(X_{r} X_{\alpha}\right)=\sum e_{\alpha_{\beta}} X_{\beta} f

\left(X_{r} X_{1}\right)=\omega_{1} X_{1} f \cdots\left(X_{r} X_{r-k}\right)=\omega_{r-k} X_{r-k} f .

\left(X_{r-k+1} X_{\alpha}\right)=\omega_{a}^{(k-1)} X_{\alpha} f .

\sum c_{r-k+2, \alpha, \beta} \omega_{\beta} X_{\beta}-\omega_{\alpha} \sum c_{\gamma-k+2, \alpha, \beta} X_{\beta}-a_{k+2} \omega_{\alpha}^{(k-1)} X_{\alpha}=0,

a_{k-2} \omega_{\alpha}^{(k-1)}=0 \text { und } c_{r-k+2, \alpha, \beta}=0 \text { für } \alpha \geqslant \beta .

a_{k-3} \omega_{\alpha}^{(k-2)}=0

\begin{aligned} & \left(X_{r} X_{\alpha_{0}}\right)=\omega_{\alpha} X_{\alpha_{0}},\left(X_{r} X_{\alpha_{1}}\right)=\omega_{\alpha} X_{\alpha_{1}}+e_{\alpha_{2} \alpha_{0}} X_{\alpha_{0}}, \ldots \\ & \left(X_{r} X_{\alpha_{x}}\right)=\omega_{\alpha} X_{\alpha_{x}}+e_{\alpha_{x} \alpha_{x-1}} X_{\alpha_{x-1}}+\cdots+e_{\alpha_{r} \alpha_{0}} X_{\alpha_{0}} . \end{aligned}

\begin{aligned} & \left(X_{e} X_{\alpha_{0}}\right)=\omega_{u}^{\prime} X_{\alpha_{0}},\left(X_{i} X_{\alpha_{1}}\right)=\omega_{u}^{\prime} X_{\alpha_{1}}+e_{\alpha_{1} \alpha_{0}}^{\prime} X_{\alpha_{0}}, \ldots \\ & \left(X_{\ell} X_{\alpha_{x}}\right)=\omega_{u}^{\prime} X_{\alpha_{x}}+e_{\alpha_{x} \alpha_{x-1}}^{\prime} X_{\alpha_{x-1}}+\cdots+e_{\alpha_{x} \alpha_{0}} X_{\alpha_{0}} f . \end{aligned}

\begin{aligned} & \left(X_{\rho} X_{\alpha_{0}}\right)=a_{\alpha} X_{\alpha_{0}}+a_{\beta} X_{\beta_{0}}+a_{\gamma} X_{\gamma_{0}}, \\ & \left(X_{0} X_{\beta_{0}}\right)=b_{a} X_{\alpha_{0}}+b_{\beta} X_{\beta_{0}}+b_{\gamma} X_{\gamma_{0}}, \\ & \left(X_{\rho} X_{\gamma_{0}}\right)=c_{\alpha} X_{\alpha_{0}}+c_{\beta} X_{\beta_{0}}+c_{\gamma} X_{\gamma_{0}} . \end{aligned}

\left(X_{\boldsymbol{\rho}} X_{\alpha_{0}}\right)=\omega_{\alpha}^{\prime} X_{\alpha_{0}},\left(X_{\boldsymbol{\rho}} X_{\beta_{0}}\right)=\omega_{p}^{\prime} X_{\beta_{0}},\left(X_{\boldsymbol{\rho}} X_{\gamma_{0}}\right)=\omega_{\gamma}^{\prime} X_{\gamma_{0}} \ldots .

\left(\sum \eta_{t} X_{t}, \sum \xi_{t} X_{t}\right)=\omega \sum \xi_{t} X_{t} t .

\left(\sum \eta_{i} X_{i}, \sum \eta_{i}^{\prime} X_{i}\right)=0 \cdots\left(\sum \eta_{i} X_{i}, \sum \eta_{i}^{(k-1)} X_{i}\right)=0

\left\{\begin{array}{l} \sum_{\iota x} \eta_{\iota} \eta_{x}^{\prime} \quad c_{i x 1}=0 \cdots \sum_{\iota x} \eta_{i} \eta_{x}^{\prime} \quad c_{i x r}=0, \\ \sum_{\iota x} \eta_{\iota} \eta_{x}^{(k-1)} c_{t x 1}=0 \cdots \sum_{\iota x} \eta_{i} \eta_{x}^{(k-1)} c_{c x r}=0 . \end{array}\right.

\left(\sum \eta_{i}^{\prime} X_{i}, \sum \eta_{i}^{\prime \prime} X_{i}\right)=0 \cdots\left(\sum \eta_{i}^{\prime} X_{i}, \sum \eta_{i}^{(k-1)} X_{i}\right)=0 \cdots

\sum_{i x} \eta_{i}^{\prime} \eta_{x}^{\prime \prime} c_{i x 1}=0 \cdots \sum_{i x} \eta_{i}^{\prime} \eta_{x}^{\prime \prime} c_{i x r}=0 .

\begin{aligned} & \left(\sum \eta_{i}^{\prime} X_{i}, \sum \xi_{i} X_{i}\right)=\omega^{\prime} \sum \xi_{i} X_{i} f \cdots \\ & \left(\sum \eta_{i}{ }^{(k-1)} X_{i}, \sum \xi_{i} X_{i}\right)=\omega^{(k-1)} \sum \xi_{i} X_{i} f . \end{aligned}

\omega^{r}-\omega^{r-1} \psi_{1}(\eta)+\omega^{r-2} \psi_{2}(\eta)-\cdots \pm \omega \psi_{r-1}(\eta)-0

\begin{aligned} & \omega^{r}-\omega^{r-1} \psi_{1}\left(\eta^{\prime}\right)+\omega^{r-2} \psi_{2}(\eta)+\cdots=0, \\ & \omega^{r}-\omega^{r-1} \psi_{1}\left(\eta^{\prime \prime}\right)+\omega^{r-2} \psi_{2}\left(\eta^{\prime \prime}\right)+\cdots=0, \\ & \omega^{r}-\omega^{r-1} \psi_{1}\left(\eta^{(i-1)}\right)+\omega^{r-2} \psi_{2}\left(\eta^{(i-1)}+\cdots=0\right. \end{aligned}

\begin{aligned} & \omega^{r}-\omega^{r-1} \psi_{1}(\eta)+\cdots=0, \\ & \omega^{r}-\omega^{r-1} \psi\left(\eta^{\prime}\right)+\cdots=0, \\ & \omega^{r}-\omega^{r-1} \psi_{1}\left(\eta^{(2-1)}\right)+\cdots=0, \end{aligned}

\tilde{\eta}_{\iota}=p_{0} \eta_{\iota}+p_{1} \eta_{i}^{\prime}+\cdots+p_{i-1} \eta_{i}{ }^{(i-1)} .

\omega^{r}-\omega^{r-1} \psi_{1}(\widetilde{\eta})+\cdots=0,

\varpi_{\alpha}=p_{0} \omega_{\alpha}+p_{1} \omega_{\alpha}^{\prime}+\cdots p_{l-1} \omega_{\alpha}^{(l-1)} .

\begin{aligned} & \left(p_{0} X_{r}+p_{1} X_{r-1}+\cdots+p_{l-1} X_{r-1+1}, X_{\alpha}\right) \\ & =\left(p_{0} \omega_{\alpha}+p_{1} \omega_{\alpha}^{\prime}+\cdots+p_{l-1} \omega_{\alpha}^{(l-1)}\right) X_{\alpha} f \end{aligned}

\begin{aligned} & \omega^{r}-\omega^{r-1} \psi_{1}(\eta)+\cdots=0, \\ & \omega^{r}-\omega^{r-1} \psi_{1}\left(\eta^{\prime}\right)+\cdots=0, \\ & \omega^{r}-\omega^{r-1} \psi_{1}\left(\eta^{(1-1)}\right)+\cdots=0 \end{aligned}

\omega^{r}-\omega^{r-1} \psi_{1}(\widetilde{\eta})+\cdots=0

\left(X_{r} X_{\alpha}\right)=\omega_{\alpha} X_{\alpha} f .

\left(X_{r-1} X_{\alpha}\right)=\omega_{\alpha}{ }^{\prime} X_{\alpha} f, \ldots\left(X_{r-2+1} X_{\alpha}\right)=\omega_{\alpha}^{(l-1)} X_{\alpha} f .

c_{\alpha \beta \gamma}\left(\omega_{\alpha}+\omega_{\beta}-\omega_{\gamma}\right)=0, \quad c_{\alpha \beta \gamma}\left(\omega_{\alpha}^{\prime}+\omega_{\beta}^{\prime}-\omega_{\gamma}^{\prime}\right)=0 \ldots,

c_{\alpha \beta \rho}\left(\omega_{\alpha}+\omega_{\beta}\right)=0, \quad c_{\alpha \beta \rho}\left(\omega_{\alpha}^{\prime}+\omega_{\beta}^{\prime}\right)=0 \ldots .

\left(X_{1} X_{i+1}\right),\left(X_{2} X_{i+2}\right) \ldots\left(X_{l} X_{2 l}\right) .

\omega_{1}+\omega_{l+1}=0, \omega_{2}+\omega_{l+2}=0 \ldots \omega_{l}+\omega_{2 l}=0 .

\left(X_{i} X_{i}^{\prime}\right)=e_{i} X_{r}+e_{i}^{\prime} X_{r-1}+\cdots+e_{i}^{(l-1)} X_{r-1+1} f ;

e_{i}^{(\nu)}+e_{i}^{(\nu)}=0 .

e_{i} \omega_{\alpha}+e_{i}^{\prime} \omega_{\alpha}^{\prime}+\cdots+e_{i}^{(i-1)} \omega_{\alpha}^{(i-1)}=-\sum\left(c_{i a o} c_{i r \alpha}-c_{i \alpha} c_{i \alpha \beta}\right) .

c_{i a 1} c_{i 1 a}, c_{i a a} c_{i 2 a}, \ldots c_{i a 1} c_{i a}, c_{i a 2} c_{i 2 a} \ldots

\omega_{\alpha_{1}}^{\nu}=\omega_{\alpha_{0}}^{\nu}+\omega_{t}^{\nu}, \omega_{\alpha_{2}}^{\nu}=\omega_{\alpha_{0}}^{\nu}+2 \omega_{t}^{\nu} \cdots \omega_{\alpha_{x}}^{\nu}=\omega_{\alpha_{0}}^{\nu}+x \omega_{i}^{\nu} .

e_{i}\left(2 \omega_{\alpha_{0}}+x \omega_{i}\right)+e_{i 2}^{\prime}\left(2 \omega_{0}^{\prime}+x \omega_{i}\right)+\cdots+e_{i}^{(l-1)}\left(2 \omega_{\alpha_{0}}^{(l-1)}+x \omega_{i}^{(l-1)}\right)=0,

\left(X_{i} X_{i^{\prime}}\right)=e_{i} X_{r} f+e_{i}^{\prime} X_{r-1} f+\cdots+e_{i}^{(i-1)} X_{r-i+1} f,

\sum_{0}^{l-1} e_{t}^{(\eta)}\left(2 \omega_{\alpha}^{(\nu)}+a_{\alpha i} \omega_{t}^{(\nu)}\right)=0

\begin{gathered} \sum e_{1}^{(\nu)} \omega_{1}^{(\nu)}=-2 E_{1}, \sum e_{2}^{(\nu)} \omega_{2}^{(\nu)}=-2 E_{2}, \ldots, \\ \sum e_{l}^{(\nu)} \omega_{l}^{(\nu)}=-2 E_{l} . \end{gathered}

\sum e_{l}^{(\nu)} \omega_{\alpha}^{(\nu)}=a_{a_{l}} E_{l} .

\omega_{\alpha}^{(\eta)}=a_{\alpha 1} E_{1} g_{1}^{(\eta)}+a_{\alpha 2} E_{2} g_{2}^{(\eta)}+\cdots+a_{a l} E_{l} g_{l}^{(\eta)} .

\left\{\begin{array}{l} \omega_{1}^{(\nu)}=a_{11} E_{1} g_{1}^{(\nu)}+a_{12} E_{2} g_{2}^{(\nu)}+\cdots+a_{11} E_{l} g_{l}^{(\nu)}, \\ \omega_{2}^{(\nu)}=a_{21} E_{1} g_{1}^{(\nu)}+a_{22} E_{2} g_{2}^{(\nu)}+\cdots+a_{21} E_{l} g_{l}^{(\nu)}, \\ \omega_{l}^{(\nu)}=a_{11} E_{1} g_{1}^{(\nu)}+a_{22} E_{2} g_{2}^{(\nu)}+\cdots+a_{u} E_{l} g_{l}^{(\nu)} . \end{array}\right.

a_{a x}=h_{a 1} a_{1 x}+h_{a 2} a_{2 x}+\cdots+h_{a l} a_{i x} \quad \text { för } \quad x=1 \ldots l .

\left\{\begin{array}{l} \omega_{\alpha}=h_{\alpha 1} \omega_{1}+h_{\alpha 2} \omega_{2}+\cdots+h_{\alpha 1} \omega_{1}, \\ \omega_{\alpha}=h_{\alpha 1} \omega_{1}^{\prime}+h_{\alpha \alpha} \omega_{2}^{\prime}+\cdots+h_{\alpha 1} \omega_{2}^{\prime}, \\ \omega_{\alpha}^{\prime-1}=h_{\alpha 1} \omega_{1}^{\prime-1}+h_{\alpha 2} \omega_{2}^{\prime-1}+\cdots+h_{\alpha i} \omega_{l}^{\prime-1} . \end{array}\right.

\begin{gathered} 2\left(e_{t} \omega_{a}+e_{t}^{\prime} \omega_{c}^{\prime}+\cdots+e_{t}^{(i-1)} \omega_{a}^{(2-1)}\right) \\ +a_{a i}\left(e_{t} \omega_{t}+e_{t}^{\prime} \omega_{t}^{\prime}+\cdots+e_{t}^{(2-1)} \omega_{i}^{(2-1)}\right)=0 \end{gathered}

m_{1} \omega_{1}+m_{2} \omega_{2}+\cdots+m_{l} \omega_{l}

-2 m_{1}+m_{2} a_{21}+\cdots+m_{l} a_{l n} .

m_{2}^{\prime}=m_{2}, m_{3}^{\prime}=m_{3}, \ldots m_{l}^{\prime}=m_{l}, m_{1}^{\prime}=-m_{1}+m_{2} a_{21}+\cdots+m_{l} a_{l 1}

\begin{gathered} m_{1}^{\prime \prime}=m_{1}^{\prime}, m_{3}^{\prime \prime}=m_{3}^{\prime}, \cdots m_{l}^{\prime \prime}=m_{l}^{\prime}, \\ m_{2}^{\prime \prime}=m_{1}^{\prime} a_{12}-m_{2}^{\prime}+m_{3}^{\prime} a_{32}+\cdots+m_{l}^{\prime} a_{i 2} . \end{gathered}

\begin{gathered} m_{1}^{(l)}=m_{1}^{(l-1)}, m_{2}^{(l)}=m_{2}^{(l-1)} \cdots m_{l-1}^{(l)}=m_{l-1}^{(l-1)}, \\ m_{l}^{(l)}=m_{1}^{(l-1)} a_{1 l}+m_{2}^{(l-1)} a_{2 l}+\cdots+m_{l-1}^{(l-1)} a_{l-1, l}-m_{l}^{(l-1)} . \end{gathered}

\begin{aligned} & \text { für } \iota=x \text { ist } b_{u}=\sum_{1}^{\infty} b_{b_{e}}^{-1} a_{e x}-1, \\ & \text { für } \iota>x \text { ist } b_{c x}=\sum_{1}^{x-1} b_{c e} a_{e x}+a_{i x}, \\ & \text { für } \iota<x \text { ist } b_{c x}=\sum_{i}^{x-1} b_{c \rho} a_{e x} . \end{aligned}

\bar{m}_{x}=\sum m_{i} b_{i x}, \quad \overline{m_{x}}=\sum \bar{m}_{i} b_{i x} \ldots

\left|\begin{array}{cccc} b_{11}-s & b_{12} & \ldots & b_{1 l} \\ b_{21} & b_{22}-s & \ldots & b_{2 l} \\ \cdot & . & . & . \\ b_{l 1} & b_{l 2} & \ldots & b_{u}-s \end{array}\right|

\left|\begin{array}{ccccc} -s-1 & a_{21} & a_{31} & \ldots & a_{l 1} \\ a_{12} s & -s-1 & a_{32} & \ldots & a_{l 2} \\ a_{13} s & a_{23} s & -s-1 & \ldots & a_{23} \\ \cdot & \cdot & \cdot & \cdot & \cdot \\ a_{11} s & a_{2 l} s & a_{3 l} s & \ldots & -s-1 \end{array}\right| .

(-1)^{t}\left\{s^{t}+M_{1} s^{s^{-1}}+M_{2} s^{l-2}+\cdots+M_{l-1} s+1\right\},

t_{1}<t_{2}<t_{3}<\cdots<t_{p} ;

\begin{aligned} & M_{1}=-l+\sum_{4 l_{2}} a_{4 l_{4}} a_{4 l_{4}}+\sum_{4 l_{4} l_{4}} a_{l_{4} l_{4}} a_{4 l_{4}} a_{4 l_{4}}+\sum_{l_{4} \cdots l_{4}} a_{4 l_{4}} a_{4 l_{4}} a_{4 l_{4} a_{4 l_{4}}} a_{4} a_{4} \\ &+\cdots+a_{12} a_{23} a_{31} \cdots a_{i-1,1} a_{11} . \end{aligned}

\begin{aligned} M_{l-1}= & -l+\sum_{l_{1} l_{2}} a_{l_{2} l_{1}} a_{l_{1} l_{2}}+\sum_{l_{2} l_{2} l_{3}} a_{l_{3} l_{4}} a_{l_{2} l_{2}} a_{l_{4} l_{4}}+\sum_{l_{4}+l_{4}} a_{l_{4} l_{4}} a_{l_{4} l_{4}} a_{l_{4} l_{4}} a_{l_{4} l_{4}} \\ + & \cdots+a_{l_{4}-1} a_{l_{1-1}, l_{2}} \cdots a_{l_{4} 3} a_{3_{2}} a_{2_{1}} a_{11} . \end{aligned}

-\left\{\left(s^{l}+1\right)+M_{1}\left(s^{l-1}+s\right)+\cdots+M_{\frac{l-1}{2}}\left(s^{\frac{l+1}{2}}+s^{\frac{l-1}{2}}\right)\right\}

\left|\begin{array}{cccc} a_{i_{1} i_{2}} & a_{i_{2} i_{2}} & \ldots & a_{i_{2} i_{\rho}} \\ a_{i_{2} i_{1}} & a_{i_{2} i_{2}} & \ldots & a_{i_{2} i_{\rho}} \\ \cdot & \cdot & \cdot & \cdot \\ a_{i_{p} i_{2}} & a_{i_{p} i_{2}} & \ldots & a_{i_{p} i_{p}} \end{array}\right|,

a_{41}=a_{42}=a_{43}=a_{51}=a_{52}=a_{53}=0

\pm \omega_{1}, \pm \omega_{2}, \pm \omega_{3}, \pm\left(\omega_{1}-\omega_{2}\right), \pm\left(\omega_{2}-\omega_{3}\right), \pm\left(\omega_{3}-\omega_{1}\right) .

\omega_{1}^{\prime}=\omega_{1}, \omega_{2}^{\prime}=\omega_{2}, \omega_{3}^{\prime}=\omega_{2}-\omega_{3},

a_{12}^{\prime}=a_{21}^{\prime}=a_{23}^{\prime}=a_{32}^{\prime}=-1, a_{13}^{\prime}=a_{31}^{\prime}=0 ;

\pm \omega_{1}^{\prime}, \pm \omega_{2}^{\prime}, \pm \omega_{3}^{\prime}, \pm\left(\omega_{1}^{\prime}-\omega_{2}^{\prime}\right), \pm\left(\omega_{2}^{\prime}-\omega_{3}^{\prime}\right) \pm\left(\omega_{1}^{\prime}-\omega_{2}^{\prime}+\omega_{3}^{\prime}\right),

\pm \omega_{1}, \pm \omega_{2}, \pm\left(\omega_{1}-\omega_{2}\right) .

\pm \omega_{1}, \pm \omega_{2}, \pm\left(\omega_{1}-\omega_{2}\right), \pm\left(2 \omega_{1}-\omega_{2}\right) .

\pm \omega_{1}, \pm \omega_{2}, \pm\left(\omega_{1}-\omega_{2}\right), \pm\left(2 \omega_{1}-\omega_{2}\right), \pm\left(3 \omega_{1}-\omega_{2}\right), \pm\left(3 \omega_{1}-2 \omega_{2}\right) .

a_{l x} a_{l l}=a_{x l} a_{l},

\begin{aligned} (-1)^{l} \Delta & =2 l-(l-1) \sum^{k} a_{l x} a_{x l}+\sum_{x k} a_{l x} a_{2 l} \\ & =2 l-(l-1)(l-1-\alpha) m+(l-\alpha-1)(l-\alpha-2) m \\ & =2 l-(l-\alpha-1)(\alpha+1) m . \end{aligned}

\omega_{\mu}-\omega_{i}, \omega_{\mu}-\omega_{i-1}-\omega_{i}, \omega_{\mu}+\omega_{\nu}-\omega_{i-1}-\omega_{i},

a_{\mu \sigma}=-1, a_{i \mu}=-1, a_{\mu a}=-2 .

\begin{gathered} \pm \omega_{1} \cdots \pm \omega_{l}, \pm\left(\omega_{\mu}-\omega_{\nu}\right), \pm\left(\omega_{\mu}-\omega_{l}\right), \pm\left(2 \omega_{l}-\omega_{\mu}\right), \\ \pm\left(2 \omega_{l}-\omega_{\mu}-\omega_{\nu}\right) . \end{gathered}

\begin{gathered} \pm \omega_{1} \cdots \pm \omega_{i}, \pm\left(\omega_{\mu}-\omega_{\nu}\right), \pm\left(\omega_{\mu}-\omega_{i}\right), \pm\left(2 \omega_{\mu}-\omega_{i}\right), \\ \pm\left(\omega_{\mu}+\omega_{\nu}-\omega_{i}\right) . \end{gathered}

a_{\mu \nu}=-1, a_{\mu, L-1}=a_{L-1, \mu}=-1, a_{\mu l}=a_{l \mu}=-1, a_{l-1, l}=a_{l, L-1}=0 .

\begin{gathered} \pm \omega_{1} \cdots \pm \omega_{l}, \pm\left(\omega_{\mu}-\omega_{\nu}\right), \pm\left(\omega_{\mu}-\omega_{l-1}\right), \pm\left(\omega_{\mu}-\omega_{l}\right), \\ \pm\left(\omega_{\mu}-\omega_{l-1}-\omega_{l}\right), \pm\left(\omega_{\mu}+\omega_{\nu}-\omega_{l-1}-\omega_{l}\right) . \end{gathered}

s^{2}+s^{n-1}+s^{n-2}+\cdots+s+1,

\begin{aligned} & \omega_{1}, \ldots \omega_{6}, \omega_{4}-\omega_{x}, \omega_{6}-\omega_{4}, \omega_{6}-\omega_{5}, \omega_{6}-\omega_{6}, \omega_{4}-\omega_{5}, \\ & -\omega_{4}+\omega_{4}+\omega_{6},-\omega_{4}+\omega_{5}+\omega_{6},-\omega_{4}-\omega_{x}+\omega_{4}+\omega_{6}, \\ & -\omega_{4}-\omega_{x}+\omega_{5}+\omega_{6},-\omega_{4}-\omega_{x}+\omega_{4}+\omega_{5}+\omega_{6}, \\ & -\omega_{1}-\omega_{2}-\omega_{3}+\omega_{4}+\omega_{5}+\omega_{6},-\omega_{1}-\omega_{2}-\omega_{3}+\omega_{4}+\omega_{5}+2 \omega_{6} . \end{aligned}

\begin{aligned} & \omega_{1} \ldots \omega_{7}, \omega_{4}-\omega_{x}, \omega_{4}-\omega_{5}, \omega_{4}-\omega_{6}, \omega_{5}-\omega_{6}, \omega_{4}-\omega_{7}, \\ & -\omega_{4}+\omega_{5}+\omega_{7},-\omega_{4}+\omega_{6}+\omega_{7},-\omega_{4}-\omega_{x}+\omega_{5}+\omega_{7}, \\ & -\omega_{4}-\omega_{x}+\omega_{6}+\omega_{7},-\omega_{1}-\omega_{x}+\omega_{5}+\omega_{6}+\omega_{7}, \\ & -\omega_{4}-\omega_{x}-\omega_{2}+\omega_{5}+\omega_{6}+\omega_{7},-\omega_{4}-\omega_{x}-\omega_{2}+\omega_{5}+\omega_{6}+2 \omega_{7}, \\ & \omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}-\omega_{5}-\omega_{6}-2 \omega_{7}, \omega_{1}+\cdots+\omega_{4}-2 \omega_{5}-\omega_{6}-2 \omega_{7}, \\ & \omega_{1}+\cdots+\omega_{4}-\omega_{5}-2 \omega_{6}-2 \omega_{7} . \end{aligned}

\begin{gathered} a_{4}=-1, a_{16}=a_{47}=a_{48}=a_{64}=a_{71}=a_{56}=-1, a_{67}=a_{76}=-1, \\ a_{65}=a_{56}=a_{75}=a_{57}=0 . \end{gathered}

\begin{aligned} & \omega_{1} \cdots \omega_{3}, \omega_{4}-\omega_{x}, \omega_{4}-\omega_{6}, \omega_{6}-\omega_{7}, \omega_{4}-\omega_{7}, \omega_{4}-\omega_{8}, \omega_{8}-\omega_{6}-\omega_{5}, \\ & \omega_{4}-\omega_{7}-\omega_{5}, \omega_{4}+\omega_{x}-\omega_{6}-\omega_{5}, \omega_{4}+\omega_{x}-\omega_{7}-\omega_{5}, \\ & \omega_{4}+\omega_{x}-\omega_{6}-\omega_{7}-\omega_{5}, \omega_{4}+\omega_{x}+\omega_{2}-\omega_{6}-\omega_{7}-\omega_{5}, \\ & \omega_{4}+\omega_{x}+\omega_{2}-\omega_{6}-\omega_{7}-2 \omega_{8}, \omega_{4}+\omega_{x}+\omega_{2}+\omega_{4}-\omega_{6}-\omega_{7}-2 \omega_{5}, \end{aligned}

s^{6}+s^{3}-s^{3}+s+1,

(s+1)\left(s^{6}-s^{3}+1\right),

s^{8}+s^{7}-s^{3}-s^{4}-s^{3}+s+1 ;

a_{x i} a_{l i}=a_{2 i} a_{l x}, \quad a_{l i-1} a_{x i}=2 a_{l-1, l} a_{l x},

\begin{aligned} (-1)^{i} \Delta & =4-2 a_{11} a_{1 i}-2 a_{i 2} a_{2 i}-\cdots-2 a_{i-2, l} a_{i, l-2}-(l-1) a_{i, l-1} a_{l-1, l} \\ & +4 a_{l-1, l}\left(a_{l i}+\cdots+a_{l, l-2}\right) . \end{aligned}

a_{14}=a_{24}=-2, a_{41}=a_{42}=a_{34}=a_{43}=-1 .

a_{41}=a_{42}=-2, a_{14}=a_{24}=a_{34}=a_{43}=-1 .

a_{l-1, l-2}=a_{l-2, l-1}=0,

a_{l l} a_{l x}=a_{l l} a_{x l} .

\begin{aligned} (-1)^{t} \Delta & =8-4 a_{1 l} a_{l 1}-\cdots-4 a_{l-3, l} a_{l, l-3}-(l-1) a_{l, l-2} a_{l-2, l} \\ & -(l-1) a_{l, l-1} a_{l-1, l}+4\left(a_{l, l-2}+a_{l, l-1}\right)\left(a_{1 l}+\cdots+a_{l-3, l}\right) \\ & -2(l-3) a_{b, l-2} a_{l-1, l} . \end{aligned}

a_{14}=a_{24}=a_{13}=a_{23}=-2 .

\begin{aligned} & \omega_{1} \ldots \omega_{4}, \omega_{1}-\omega_{2}, \omega_{1}-\omega_{3}, \omega_{1}-2 \omega_{3}, \omega_{1}-\omega_{4}, \omega_{1}-2 \omega_{4}, \\ & \omega_{2}-\omega_{3}, \omega_{2}-2 \omega_{3}, \omega_{2}-\omega_{4}, \omega_{2}-2 \omega_{4}, \omega_{3}-\omega_{4}, \omega_{1}-\omega_{3}-\omega_{4}, \end{aligned}

\begin{aligned} & \omega_{2}-\omega_{3}-\omega_{4}, \omega_{1}+\omega_{2}-2 \omega_{3}, \omega_{1}+\omega_{2}-2 \omega_{4}, \omega_{1}+\omega_{2}-\omega_{3}-\omega_{4}, \\ & \omega_{1}+\omega_{2}-2 \omega_{3}-\omega_{4}, \omega_{1}+\omega_{2}-\omega_{3}-2 \omega_{4}, 2 \omega_{1}+\omega_{2}-2 \omega_{3}-2 \omega_{4}, \\ & \omega_{1}+2 \omega_{2}-2 \omega_{3}-2 \omega_{4} . \end{aligned}

\begin{aligned} & \omega_{1} \cdot \omega_{4}, \omega_{1}-\omega_{2}, \omega_{1}-\omega_{3}, 2 \omega_{1}-\omega_{3}, \omega_{1}-\omega_{4}, 2 \omega_{1}-\omega_{4}, \\ & \omega_{2}-\omega_{3}, 2 \omega_{2}-\omega_{3}, \omega_{2}-\omega_{4}, 2 \omega_{2}-\omega_{4}, \omega_{3}-\omega_{4}, \omega_{1}+\omega_{2}-\omega_{3}, \\ & \omega_{1}+\omega_{2}-\omega_{4}, 2 \omega_{1}-\omega_{3}-\omega_{4}, 2 \omega_{2}-\omega_{3}-\omega_{4}, \omega_{1}+\omega_{2}-\omega_{3}-\omega_{4}, \\ & 2 \omega_{1}+\omega_{2}-\omega_{3}-\omega_{4}, \omega_{1}+2 \omega_{2}-\omega_{3}-\omega_{4}, 2 \omega_{1}+2 \omega_{2}-\omega_{3}-\omega_{4}, \\ & 2 \omega_{1}+2 \omega_{2}-2 \omega_{3}-\omega_{4}, 2 \omega_{1}+2 \omega_{2}-\omega_{3}-2 \omega_{4} . \end{aligned}

a_{27}=-1, a_{37}=\cdots=a_{67}=0 ;

a_{27}=a_{37}=0, a_{47}=a_{57}=1, a_{67}=-1 ;

a_{27}=0, a_{37}=a_{47}=a_{57}=-1, a_{67}=0 ;

a_{27}=a_{37}=-1, a_{47}=a_{57}=0, a_{67}=-1 ;

a_{27}=a_{37}=a_{47}=a_{57}=0, a_{67}=-1 .

a_{2 \mathrm{~S}}=a_{38}=a_{48}=0, a_{58}=a_{68}=1, a_{78}=-1,

a_{28}=0, a_{38}=a_{48}=a_{58}=a_{68}=-1, a_{78}=0,

a_{28}=a_{38}=-1, a_{48}=a_{55}=a_{68}=0, a_{75}=-1,

a_{23}=a_{38}=0, a_{48}=-1, a_{58}=a_{68}=0, a_{78}=-1,

a_{28}=a_{38}=a_{48}=-1, a_{58}=a_{68}=-1, a_{78}=0 .

c_{\alpha \beta \gamma}\left(\omega_{\alpha}+\omega_{\beta}-\omega_{\gamma}\right)=0, \quad c_{\alpha \beta \varphi}\left(\omega_{\alpha}+\omega_{\beta}\right)=0,

\sum_{0}^{L-1} e_{i}^{(\gamma)} \omega_{\alpha}^{(\gamma)}=\sum\left(c_{i+\alpha} c_{i \alpha}-c_{i \alpha} c_{i^{\prime}, \alpha}\right) .

\sum e_{i}^{(\eta)} \omega_{i}^{(\eta)}=-2 E_{i}

\left.a_{a t} E_{t}=\sum c_{t, \alpha} c_{t^{\prime} \alpha^{2}}-c_{t a *} c_{t^{\prime} \alpha \alpha}\right) .

=\sum_{v}^{h-1} e_{t}^{(v)} X_{r-v} f .

\pm\left(\omega_{\imath}-\omega_{x}\right), \cdots \pm\left(\omega_{\imath}+a_{i x} \omega_{x}\right)

\omega_{t}-2 \omega_{x} \cdots \omega_{t}+a_{t x} \omega_{x} .

\begin{array}{ll} \left(X_{i x} X_{x 2}\right)=-X_{i 2}+\delta_{i x} X_{x x}, & \left(X_{x 0} X_{0 i}\right)=-X_{x i}-\delta_{x i} \sum X_{\mu \mu}, \\ \left(X_{0 x} X_{x 2}\right)=-X_{02}, & \left(X_{i x} X_{x 0}\right)=-X_{i 0} . \end{array}

X_{r} f=\sum \eta_{\imath} X_{\imath \imath} f, \quad X_{r-1} f=\sum \eta_{\imath}^{\prime} X_{\imath \imath} f \text { u. s. w. }

\begin{aligned} \alpha & =1 \cdots l-2: \omega_{\alpha}=\pi_{\alpha}+\pi_{l} \\ \omega_{l-1} & =\pi_{l-1}+\pi_{l}, \quad \omega_{l}=-\pi_{l+1}+\pi_{l} . \end{aligned}

\eta_{12} X_{12}+\eta_{34} X_{34}+\cdots+\eta_{21-1,21} X_{21-1,2 t}

\left(X_{\alpha \beta} X_{\alpha \gamma}\right)=X_{\beta \gamma} f, \quad\left(X_{\alpha \beta} X_{\gamma \delta}\right)=0 .

\eta_{12} X_{12}+\eta_{34} X_{34}+\cdots+\eta_{2 i-1,2 i} X_{2 i-1,2 i} f

X_{1} \ldots X_{i}, X_{1} \ldots X_{i}, X_{\frac{i+x}{2}}, X_{\frac{i-x}{2}}, X_{\frac{-i+x}{2}}, X_{\frac{-i-x}{2}} .

\begin{gathered} \sum i e_{i}^{v} \omega_{i}^{v}=-\sum e_{-i}^{v} \omega_{i}^{v}=-2 E_{i}=-2 E_{-i}--2 g_{i}^{2}, \\ \sum e_{\frac{i+x}{2}}^{v} \omega_{\frac{i+x}{2}}^{v}=\frac{1}{2} \sum e_{\frac{4+x}{2}}^{v}\left(\omega_{i}^{2}+\omega_{x}^{2}\right)=-\sum \frac{e^{v}-x}{2}\left(\omega_{i}^{v}+\omega_{x}^{\prime}\right) \\ =-2 E_{\frac{4+x}{2}}=-2 E_{\frac{-t-x}{2}}=-2 g_{\frac{i+x}{2}}^{2} \end{gathered}

\sum e_{i}^{r} \omega_{x}^{v}=0, \quad \sum e_{\frac{i+x}{2}}^{y} \omega_{\frac{i-x}{2}}^{v}=0, \quad \sum_{\frac{t+x}{2}}^{y} \omega_{z}^{v}=0,

\frac{\frac{e_{t+x}^{y}}{2}}{\frac{E_{t+x}^{y}}{2}}=\frac{e_{t}^{y}}{E_{t}}+\frac{e_{x}^{y}}{E_{x}}, \quad \frac{\frac{e_{t-x}^{y}}{2}}{\frac{E_{t-x}}{2}}=\frac{e_{t}^{y}}{E_{t}}-\frac{e_{x}^{y}}{E_{x}} .

\begin{array}{rl} \quad\left(i,-i, \frac{-i+x}{2}\right), & \left(i,-i, \frac{i-x}{2}\right), \\ \left(\frac{-i+x}{2}, \frac{i-x}{2},-i\right), & \left(\frac{i+x}{2}, \frac{-i-x}{2}, i\right), \\ 2 & i+x \end{array}

\begin{aligned} & c \frac{-t+x i+x}{2} \quad c_{\frac{t+x}{2}-t-t+x}^{2}=c_{-t} \frac{t-x}{2} \frac{-t-x}{2} \frac{c-t-x}{2} \iota \frac{t-x}{2}=g_{t}^{2}, \\ & \frac{c_{-t+x}}{2} \iota \frac{t+x}{2} \quad c_{\frac{t+x}{2} \frac{t-x}{2} \iota}=c_{\frac{t-x}{2}-\iota \frac{-t-x}{2}} \frac{c_{-t-x}}{2} \frac{-t+x}{2}-\iota=2 g_{t-x}^{2}, \\ & \frac{c_{t+x}}{2}, \frac{-t+x}{2} \quad \frac{c-t+x}{2} \frac{-t-x}{2}=\frac{c_{-t-x}}{2}, \frac{t-x}{2} \quad c_{\frac{t-x}{2} \frac{t+x}{2} \iota}^{2}=2 g_{\frac{t+x}{2}}^{2} . \end{aligned}

c_{\frac{4+x}{2}} \frac{-4+2}{2} \frac{x+2}{2} c_{\frac{x+2}{2}} \frac{-1-x}{2} \frac{-1+2}{2}=g_{\frac{1+x}{2}}^{2} .

c_{t} \frac{-i+x}{2} \frac{i+x}{2} c_{\frac{1+x}{2}}=\frac{-i+2}{2} \frac{x+2}{2}=c_{1}=\frac{-4+2}{2} \frac{1+2}{2} c_{1+2}=\frac{4+x}{2} \frac{x+2}{x}

{ }_{t}^{c} \frac{-4+x}{2} \frac{4 x}{2} c_{\frac{4 x}{2}} \frac{-x+2}{2} \frac{4+2}{2}=c_{3} \frac{-4+2}{2} \frac{4+2}{2} c_{-4+\frac{x}{2}}=\frac{-x+2}{2}=\frac{4+4}{2}

c_{\frac{4+2}{2}-x+2 \frac{1-x}{2}} c_{\frac{4-x}{2}} \frac{4+x}{2}, \quad c_{4+x} \frac{-x-2}{2} \frac{1-2}{2} c_{4-2} \frac{4+2}{2}

c_{\frac{1+x}{2}}=\frac{-+\lambda}{2} \frac{x+2}{2} c_{\frac{x+2}{2}} \frac{-x+\mu}{2} \frac{2+\mu}{2}+c_{\frac{-x+\mu}{2}}^{\frac{4+x}{2}} \frac{4+\mu}{2} c_{\frac{4+\mu}{2}}=\frac{-4+2}{2} \frac{2+\mu}{2}=0 .

c_{\frac{t+x}{2}-\frac{t+2}{2} \frac{x+2}{2}}=\sqrt{\frac{p_{x 2} p_{2 x} p_{t 2}}{p_{t x} p_{x i} p_{2 i}}} \frac{\frac{g_{t+x}}{2} \frac{g_{t-2}}{2}}{\frac{g_{x+2}}{2}} .

\sqrt{p_{i,-x}}=i \sqrt{p_{i x}}, \quad \sqrt{p_{-i, x}}=\frac{-i}{\sqrt{p_{i x}}}, \quad \sqrt{p_{-i,-x}}=\frac{1}{\sqrt{p_{i x}}} .

\begin{aligned} & \left(X_{i x} X_{x \lambda}\right)=-X_{i \lambda}+\varepsilon_{i \lambda} X_{x x}, \\ & \left(X_{x 0} X_{02}\right)=-X_{x \lambda}-\varepsilon_{x \lambda} \sum X_{\mu \mu}, \\ & \left(X_{0} X_{x \lambda}\right)=-X_{0 \lambda}, \quad\left(X_{i \lambda} X_{x 0}\right)=-X_{i 0} \end{aligned}

\left(X_{i x} X_{i a}\right)=X_{x a} f, \quad\left(X_{i x} X_{i_{\mu} \mu}\right)=0 .

X_{r} f, X_{r-1} f \ldots X_{r-i+1} f, X_{\imath} f, X_{-\iota} f, X_{\frac{c+x}{2}} f, \frac{X_{t-x}}{2} f, X_{\frac{-t-x}{2}} f

\begin{gathered} \left(X_{r} X_{i}\right)=\omega_{i} X_{i} \quad\left(X_{r} X_{-i}\right)=-\omega_{i} X_{i} \\ \left(X_{r} X_{i+x}\right)=\frac{1}{2}\left(\omega_{i}+\omega_{x}\right) X_{i+x} f \text { u. s. w. } \\ \left(X_{r-1} X_{i}\right)=\omega_{i}^{\prime} X_{i} \quad\left(X_{r-1} X_{x-i}\right)=-\omega_{i}^{\prime} X_{i \prime} \\ \left(X_{r-1} X_{i+x}\right)=\frac{1}{2}\left(\omega_{i}^{\prime}+\omega_{x}^{\prime}\right) X_{i+x} f \cdots \\ \left(X_{i} X_{-i}\right)=e_{i} X_{i}+e_{i}^{\prime} X_{r-1}+\cdots \\ \left(X_{i+x} X_{i-x}^{\prime}\right)=e_{\frac{i+x}{2}} X_{r}+e_{i+x}^{\prime} X_{r-1}+\cdots \end{gathered}

\sum e_{i}^{v} \omega_{i}^{v}=-2 g_{i}^{2}, \quad \frac{1}{2} \sum v_{i+x}^{v}\left(\omega_{i}^{v}+\omega_{x}^{v}\right)=-2 g_{i+x}^{2},

\begin{gathered} \frac{e_{i+x}^{y}}{z} \\ g_{i+x}^{2} \\ z \end{gathered}=\frac{e_{i}^{y}}{g_{i}^{2}}+\frac{e_{x}^{y}}{g_{x}^{2}} \quad \text { u. s. w. }

V p_{i,-x}=i \sqrt{p_{i x}}, \quad \sqrt{p_{-i, x}}=\frac{-i}{V p_{i x}},

\begin{aligned} & c_{i-i+x i+x}=p_{i x} \frac{g_{i} g_{i-x}}{\frac{g_{i+x}}{2}}, \quad c_{\frac{i-x i+x}{2} \frac{i}{2}}=\frac{2}{p_{i x}}=\frac{g_{i+x} g_{i-x}}{g_{i}}, \\ & c_{\frac{i+x-i+2 x+2}{2}}=\frac{\sqrt{p_{x i}} \sqrt{p_{i x}} \sqrt{p_{i, 2}}}{\sqrt{p_{i x}} \sqrt{p_{x i}} \sqrt{p_{i i}}} \frac{g_{i+x} g_{i-2}}{\frac{2}{g_{x+2}}} . \end{aligned}

\pm \omega_{1}, \pm \omega_{2}, \pm\left(\omega_{1}-\omega_{2}\right), \pm\left(2 \omega_{1}-\omega_{2}\right), \pm\left(3 \omega_{1}-\omega_{2}\right), \pm\left(3 \omega_{1}-2 \omega_{2}\right) .

\omega_{1}, \omega_{2}, \omega_{1}-\omega_{2}, \omega_{1}+\omega_{2}, 2 \omega_{1}-\omega_{2}, \omega_{1}-2 \omega_{2},

\begin{array}{lll} e_{1} \omega_{1}+e_{1}^{\prime} \omega_{1}^{\prime}=-2 E_{1}, & e_{2} \omega_{1}+e_{2}^{\prime} \omega_{1}^{\prime}=-E_{2}, & e_{3} \omega_{1}+e_{3}^{\prime} \omega_{1}^{\prime}=-E_{3}, \\ e_{1} \omega_{2}+e_{1}^{\prime} \omega_{2}^{\prime}=-E_{1}, & e_{2} \omega_{2}+e_{2}^{\prime} \omega_{2}^{\prime}=-2 E_{2}, & e_{3} \omega_{2}+e_{3}^{\prime} \omega_{2}^{\prime}=E_{3}, \\ e_{4} \omega_{1}+e_{4}^{\prime} \omega_{1}^{\prime}=-E_{4}, & e_{5} \omega_{1}+e_{5}^{\prime} \omega_{1}^{\prime}=-E_{5}, & e_{6} \omega_{1}+e_{6}^{\prime} \omega_{1}^{\prime}=0, \\ e_{4} \omega_{2}+e_{4}^{\prime} \omega_{2}^{\prime}=-E_{4}, & e_{5} \omega_{2}+e_{5}^{\prime} \omega_{2}^{\prime}=0, & e_{6} \omega_{2}+e_{6}^{\prime} \omega_{2}^{\prime}=E_{6} . \end{array}

\begin{aligned} \frac{e_{2}}{E_{3}}=\frac{e_{1}}{E_{1}}-\frac{e_{2}}{E_{2}}, & \frac{e_{4}}{E_{1}}=\frac{1}{3}\left(\frac{e_{1}}{E_{1}}+\frac{e_{2}}{E_{2}}\right), \quad \frac{e_{3}}{E_{5}}=\frac{1}{3}\left(\frac{2 e_{1}}{E_{1}}-\frac{e_{2}}{E_{2}}\right) . \\ & \frac{e_{6}}{E_{6}}=\frac{1}{3}\left(\frac{e_{1}}{E_{1}}-2 \frac{e_{6}}{E_{2}}\right) \end{aligned}

\begin{aligned} & c_{124}=3 p_{1} \frac{g_{1} g_{2}}{g_{4}}, \quad c_{124}=\frac{3}{p_{1}} \frac{g_{1} g_{2}}{g_{4}}, \quad c_{241}=p_{1} \frac{g_{2} g_{4}}{g_{1}}, \quad c_{241}=\frac{1}{p_{1}} \frac{g_{2} g_{4}}{g_{1}}, \\ & c_{422}=\frac{1}{p_{1}} \frac{g_{1} g_{4}}{g_{2}}, \quad c_{412}=p_{1} \frac{g_{1} g_{1}}{g_{2}}, \quad c_{135}=3 p_{2} \frac{g_{1} g_{3}}{g_{3}}, \quad c_{155}=\frac{3}{p_{2}} \frac{g_{1} g_{3}}{g_{3}}, \\ & c_{515}=p_{2} \frac{g_{1} g_{3}}{g_{3}}, \quad c_{515}=\frac{1}{p_{2}} \frac{g_{1} g_{5}}{g_{3}}, \quad c_{551}=\frac{1}{p_{2}} \frac{g_{3} g_{5}}{g_{1}}, \quad c_{551}=p_{2} \frac{g_{3} g_{5}}{g_{1}}, \\ & c_{231}=2 p_{3} \frac{g_{2} g_{3}}{g_{1}}, \quad c_{251}=\frac{2}{p_{3}} g_{2} g_{3}, \quad c_{12^{\prime} 3}=\frac{2}{p_{3}} \quad g_{3} \quad g_{2}, \quad c_{123}=2 p_{3} \quad g_{1} g_{2}, \\ & c_{s 12}=\frac{2}{p_{s}} \frac{g_{1} g_{3}}{g_{2}}, \quad c_{312}=2 p_{3} \frac{g_{1} g_{3}}{g_{2}}, \quad c_{263}=p_{4} \frac{g_{2} g_{6}}{g_{3}}, \quad c_{265}=\frac{1}{p_{4}} \frac{g_{2} g_{3}}{g_{3}}, \\ & c_{\zeta^{2} \zeta^{3}}=3 p_{4} \frac{g_{2} g_{3}}{g_{6}}, \quad c_{3 \zeta^{\prime}}=\frac{3}{p_{4}} \frac{g_{2} g_{3}}{g_{6}}, \quad c_{632}=p_{4} \frac{g_{2} g_{6}}{g_{2}}, \quad c_{632}=\frac{1}{p_{1}} \frac{g_{2} g_{6}}{g_{2}}, \\ & c_{46}=p_{3} \frac{g_{4} g_{6}}{g_{3}}, c_{463}=\frac{1}{p_{3}} \frac{g_{4} g_{5}}{g_{5}}, c_{56}=\frac{1}{p_{3}} \frac{g_{3} g_{4}}{g_{6}}, c_{546}=p_{5} \frac{g_{5} g_{6}}{g_{6}}, \\ & c_{654}=p_{5} \frac{g_{6} g_{3}}{g_{4}}, \quad c_{634}=\frac{1}{p_{5}} \frac{g_{6} g_{5}}{g_{4}} . \end{aligned}

p_{1} p_{5}=p_{2} p_{4} .

\begin{gathered} \left(X_{1} X_{1}^{\prime}\right)=2 X_{7} f+X_{8} f, \quad\left(X_{2} X_{7}\right)=X_{7} f+2 X_{3} f ; \\ \omega_{1}=-1, \quad \omega_{1}^{\prime}=0, \quad \omega_{2}=0, \quad \omega_{2}^{\prime}=-1, \quad E_{1}=E_{2}=1 . \end{gathered}

E_{3}=E_{4}=E_{5}=E_{6}=1, \quad p_{1}=p_{2}=p_{3}=p_{4}=1 .

\begin{array}{lll} \left(X_{7} X_{1}\right)=-X_{1} f, & \left(X_{7} X_{1}\right)=X_{1} f, & \left(X_{8} X_{1}\right)=0, \end{array} \quad \begin{array}{ll} \left(X_{8} X_{1}\right)=0, \\ \left(X_{7} X_{2}\right)=0, & \left(X_{7} X_{2}\right)=0, \end{array} \quad\left(X_{8} X_{2}\right)=-X_{8} f, \quad\left(X_{8} X_{2}\right)=X_{2} f .

\pm \omega_{i}, \quad \pm \omega_{i} \pm \omega_{x}, \quad \frac{ \pm \omega_{1} \pm \omega_{2} \pm \omega_{s} \pm \omega_{4}}{2} .

X_{r}, X_{r-1}, X_{r-2}, X_{r-3} ; X_{ \pm i}, X_{ \pm \pm x}, X_{ \pm \frac{1 \pm 2 \pm 3 \pm 4}{2}},

\begin{aligned} & c_{-1} \frac{1+2+3+4}{2} \frac{-1+2+3+4}{2} c_{-1+2+3+4} \frac{1+2+3+4}{2}=g_{1}{ }^{2}, \\ & c_{-1-2-3-4} \frac{1-2-3-4}{2} c_{\frac{1-2-3-4}{2} \frac{1+2+3+4}{2}}=g_{\frac{1+2+3+4}{2}}^{2} \end{aligned}

\begin{gathered} c_{-1} \frac{1+2+3+4}{2} \frac{-1+2+3+4}{2}, \quad c_{-1} \frac{1-2+3+4}{2} \frac{-1-2+3+4}{2}, \quad c_{-1} \frac{1+2-3+4}{2} \frac{-1+2-3+4}{2}, \\ c_{-1} \frac{1+2+3-4}{2} \frac{-1+2+3-4}{2} \end{gathered}

\left(1,-1+2, \frac{1-2+3+4}{2}\right) \text { and }\left(1,-1+2, \frac{-1-2+3+4}{2}\right),

\left\{\begin{array}{c} c_{1,-1+2,2} c_{2} \frac{1-2+3+4}{2} \frac{1+2+3+4}{2} \\ +c_{-1+2, \frac{1-2+3+4}{2}-1+2+3+4}^{2} \frac{c_{-1+2+3+4}}{2} \frac{1+2+3+4}{2}=0, \\ c_{1,-1+2,2} c_{2} \frac{-1-2+3+4}{2} \frac{-1+2+3+4}{2} \\ +c_{\frac{-1-2+3+4}{2}} \frac{1-2+3+4}{2} \frac{c_{1-2+3+4}}{2},-1+2, \frac{-1+2+3+4}{2}=0, \end{array}\right.

\begin{gathered} c_{1} \frac{-1+2+3+4}{2} \frac{1+2+3+4}{2} c_{2} \frac{-1-2+4+4}{2} \frac{-1+2+3+4}{2} \\ +c_{1} \frac{-1-2+4+1}{2} \frac{1-2+3+1}{2} c_{2} \frac{1-2+3+1}{2} \frac{1+2+3+4}{2}=0 . \end{gathered}

\begin{gathered} C \frac{1+2+3+4}{2} \frac{1-2+3+4}{2}, C{ }_{-2} \frac{1+2-3+4}{2} \frac{1-2-3+4}{2}, C \\ C-\frac{1+2+3+1}{2} \frac{1+2+3+3+1}{2} \end{gathered}

\left(\frac{1+2+3+4}{2}, \frac{-1-2-3-4}{2}, 1+2\right)

\pm \omega_{1}, \quad \pm \omega_{2} \pm \omega_{x}, \quad \pm \omega_{1} \pm \omega_{2} \pm \omega_{2} \pm \omega_{4} .

\begin{gathered} \left(1,-1, \frac{-1+2+3+4}{2}\right),\left(\frac{1+2+3+4}{2}, \frac{-1-2-3-4}{2}, 1\right), \\ \left(1,2, \frac{-1-2+3+4}{2}\right) \end{gathered}

c_{-1}^{c} \frac{1+2+3+4}{2} \frac{-1+2+3+4}{8}

\omega_{6}, \pi_{i}+\pi_{x}-\omega_{i}, \pi_{i}+\pi_{x}+\pi_{i}+\pi_{\mu}-\omega_{G}

\begin{aligned} & \pi_{1}+\pi_{x}, \quad \pi_{i}-\pi_{x}, \quad \omega_{7}, \quad \pi_{i}+\pi_{x}-\omega_{7}, \quad \pi_{i}+\pi_{x}+\pi_{\lambda}+\pi_{\mu}-\omega_{7}, \\ & \pi_{1}+\pi_{2}+\pi_{3}+\pi_{4}+\pi_{5}+\pi_{6}-\omega_{7}, \quad \pi_{1}+\pi_{2}+\pi_{3}+\pi_{4}+\pi_{5}+\pi_{6}-2 \omega_{7} . \end{aligned}

\begin{gathered} \pi_{i}+\pi_{x}, \pi_{i}-\pi_{x}, \omega_{8}, \pi_{i}+\pi_{x}-\omega_{8}, \pi_{i}+\pi_{x}+\pi_{2}+\pi_{\mu}-\omega_{5}, \\ \pi_{i_{1}}+\cdots+\pi_{i_{2}}-\omega_{8}, \pi_{i_{1}}+\cdots+\pi_{i_{6}}-2 \omega_{8}, \\ 2 \pi_{i_{4}}+\pi_{i_{2}}+\pi_{i_{2}}+\pi_{i_{4}}+\pi_{i_{3}}+\pi_{i_{6}}+\pi_{i_{7}}-2 \omega_{8}, \end{gathered}