possibly zero, by our convention) if $j$ is even (resp. odd).

\[t411\] The morphism $$\psi=\prod_j \psi_j : \tilde{G} \longrightarrow (\mathbb{Z}/2\mathbb{Z})^{\beta}$$ is surjective.

Moreover, the morphism $$\varphi \times \psi : \tilde{G} \longrightarrow  \prod_{i:even} \mathrm{O}(B_i/Z_i, \bar{q}_i)_{\mathrm{red}} \times \prod_{i:odd} \mathrm{Sp}(B_i/Y_i, h_i)\times (\mathbb{Z}/2\mathbb{Z})^{\beta}$$ is also surjective.

We first show that $\psi_j$ is surjective. Recall that for such an integer $j$, $L_j$ is *of type I* and $L_{j+2}, L_{j+3}, L_{j+4}$ (resp. $L_{j-1}, L_{j+1},$ $L_{j+2},  L_{j+3}$) are *of type II* if $j$ is even (resp. odd). We define the closed subgroup scheme $F_j$ of $\tilde{G}$ defined by the following equations:

-   $m_{i,k}=0$ *if $i\neq k$*;

-   $m_{i,i}=\mathrm{id}, z_i^{\ast}=0, m_{i,i}^{\ast}=0, m_{i,i}^{\ast\ast}=0$ *if $i\neq j$*;

-   and for $m_{j,j}$, $$\left \{
      \begin{array}{l l}
      s_j=\mathrm{id~}, y_j=0, v_j=0, z_j=\pi z_j^{\ast} & \quad  \textit{if $i$ is even and $L_i$ is \textit{of type} $\textit{I}^o$};\\
      s_j=\mathrm{id~}, r_j=t_j=y_j=v_j=u_j=w_j=0, z_j=\pi z_j^{\ast}  & \quad \textit{if $i$ is even and  $L_i$ is \textit{of type} $\textit{I}^e$};\\
      s_j=\mathrm{id~}, r_j=t_j=y_j=v_j=u_j=w_j=0 & \quad \textit{if $i$ is odd and  $L_i$ is \textit{free of type I}}.\\
        \end{array} \right.$$

A formal matrix form of an element of $F_j(R)$ for a $\kappa$-algebra $R$ is then $$\begin{pmatrix} id&0& & \ldots& & &0\\ 0&\ddots&& & & &\\ & &id& & & & \\  \vdots & & &m_{j,j} & & &\vdots
\\ & & & & id & & \\  & & & & &\ddots &0 \\ 0& & &\ldots & &0 &id \end{pmatrix}$$
such that $$m_{j,j}=\left\{
\begin{array}{l l}
\begin{pmatrix}id&0\\0&1+2 z_j^{\ast} \end{pmatrix} & \quad \textit{if $j$ is even and $L_j$ is of type $I^o$};\\
\begin{pmatrix}id&0&0\\0&1+\pi x_j&2 z_j^{\ast}\\0&0&1 \end{pmatrix} & \quad \textit{if $j$ is even and  $L_j$ is of type $I^e$};\\
\begin{pmatrix}id&0&0\\0&1+\pi x_j&0\\0&\pi z_j&1 \end{pmatrix} & \quad \textit{if $j$ is odd and  