\begin{aligned}
&\begin{aligned}(\tau\lambda)\psi(a)(\lambda^{-1}\tau)(X,Y,\xi,\eta)=(\tau\lambda)\psi(a)(-\tau Y,\tau X,-\tau\eta,\tau\xi)\end{aligned} \\
&=(\tau\lambda)\bigg(\begin{pmatrix}-a\tau\eta_1&-\tau y_3&-\tau\overline{y}_2\\-\tau\overline{y}_3&-a^{-1}\tau\eta_2&-a^{-1}\tau y_1\\-\tau y_2&-a^{-1}\tau\overline{y}_1&-a^{-1}\tau\eta_3\end{pmatrix},\begin{pmatrix}a^{-1}\tau\xi_1&\tau x_3&\tau\overline{x}_2\\\tau\overline{x}_3&a\tau\xi_2&a\tau x_1\\\tau x_2&a\tau\overline{x}_1&a\tau\xi_3\end{pmatrix},-a\tau\eta,a^{-1}\tau\xi\bigg) \\
&\left.=\left(\begin{pmatrix}\tau a^{-1}\xi_1&x_3&\overline{x}_2\\\overline{x}_3&\tau a\xi_2&\tau ax_1\\x_2&\tau a\overline{x}_1&\tau a\xi_3\end{pmatrix}\right.,\begin{pmatrix}\tau a\eta_1&y_3&\overline{y}_2\\\overline{y}_3&\tau a^{-1}\eta_2&\tau a^{-1}y_1\\y_2&\tau a^{-1}\overline{y}_1&\tau a^{-1}\eta_3\end{pmatrix},\tau a^{-1}\xi,\tau a\eta\right) \\
&=\psi(\tau a^{-1}).
\end{aligned}

\begin{aligned}
&\begin{aligned}-L_{X_{13}}&=\left(\frac{1}{2}\sin\alpha\cos\beta\sin2\gamma+\cos\alpha\tan\beta\sin^2\gamma-\frac{1}{2}\sin\alpha\sin\beta\tan\beta\sin2\gamma\right)\frac{\partial}{\partial\alpha}\end{aligned} \\
&\begin{aligned}+\left(\frac12\cos\alpha\sin\beta\sin2\gamma-\sin\alpha\sin^2\beta\cos^2\gamma-\sin\alpha\cos^2\beta\sin^2\gamma\right)\frac\partial{\partial\beta}\end{aligned} \\
&\begin{aligned}+\left(\frac14\sin\alpha\sin2\beta\sin2\gamma-\frac12\sin\alpha\tan\beta\sin2\gamma+\cos\alpha\sec\beta\sin^2\gamma\right)\frac{\partial}{\partial\gamma}\end{aligned} \\
&+\left(\left(\frac12\sin\alpha\sin2\beta\cos^2\gamma+\frac12\sin\alpha\sin2\beta-\frac12\cos\alpha\cos\beta\sin2\gamma\right)z_{12}\right.  \\
&+(\sin\alpha\cos2\beta\cos\gamma+\cos\alpha\sin\beta\sin\gamma)\biggr)\frac{\partial}{\partial z_{12}} \\
&+\left(\left(\frac12\sin\alpha\sin2\beta\cos2\gamma-\cos\alpha\cos\beta\sin2\gamma\right)z_{13}+(\sin\alpha\cos2\beta\cos\gamma\right.  \\
&\left.\left.+\cos\alpha\sin\beta\sin\gamma\right)z_{23}+\left(\frac12\sin\alpha\sin2\beta\sin2\gamma+\cos\alpha\cos\beta\cos2\gamma\right)\right)\frac{\partial}{\partial z_{13}} \\
&+\left(\left(-\frac12\sin\alpha\sin2\beta-\frac12\sin\alpha\sin2\beta\sin^2\gamma-\frac12\cos\alpha\cos\beta\sin2\gamma\right)z_{23}\right. \\
&+(\sin\alpha\cos2\beta\sin\gamma-\cos\alpha\sin\beta\cos\gamma)\Bigg)\frac{\partial}{\partial z_{23}}.
\end{aligned}

\begin{aligned}
&\sum_S(-1)^{|S|}\frac{1-\prod_{i\notin S}\left(\frac{X_i(1+X_i)}{Q+X_i}\right)^{m+1}}{1-\prod_{i\notin S}\frac{X_i(1+X_i)}{Q+X_i}}\prod_iX_i \\
&\times\prod_{i\in S}X_{i}^{m+n-1}(1+X_{i})^{m+1}(Q+X_{i})^{-m}(X_{i}+r+Q)^{n-1} \\
&\times\prod_{i\notin S}(1+X_i)(Q+rX_i+QX_i)^{n-1} \\
&&\times\prod_{1\leq i<j\leq n,\{i,j\}\cap S\neq\emptyset}\left(\frac{Y_j(1+Y_j)}{Q+rY_j+QY_j}-\frac{Y_i(1+Y_i)}{Q+rY_i+QY_i}\right) \\
&&&\times\sum_{k\notin S}(Q-X_{k}^{2})X_{k}^{-1}(1+X_{k})^{-1} \\
&&&\times\prod_{\overset{1\leq i\leq k-1}{i\notin S}}\frac{(Q+(Q+r)X_k+X_i+X_iX_k)(X_iX_k-Q)}{(Q+rX_k+QX_k)(Q+rX_i+QX_i)} \\
&&&\times\prod_{\overset{k+1\leq i\leq n}{i\notin S}}\frac{(Q+(Q+r)X_k+X_i+X_iX_k)(Q-X_iX_k)}{(Q+rX_k+QX_k)(Q+rX_i+QX_i)} \\
&&&&\times\prod_{1\leq i<j\leq n,i,j\notin S\cup\{k\}}\left(\frac{X_j(1+X_j)}{Q+rX_j+QX_j}-\frac{X_i(1+X_i)}{Q+rX_i+QX_i}\right).
\end{aligned}