tent group by a theorem of Lazard which is stated at the beginning of Appendix \[App:AppendixA\].

Recall that we have defined the morphism $\varphi$ in Section \[red\]. The morphism $\varphi$ extends to an obvious morphism $$\tilde{\varphi} : \tilde{M} \longrightarrow  \prod_{i:even}\mathrm{GL}_{\kappa}(B_i/Z_i) \times  \prod_{i:odd}\mathrm{GL}_{\kappa}(B_i/Y_i)$$ such that $\tilde{\varphi}|_{\tilde{G}}=\varphi $. Note that $Y_i\otimes_AR$, when $i$ is odd, is preserved by an element of $\underline{M}(R)$ for a flat $A$-algebra $R$ (cf. Lemma \[l42\]). By using this, the construction of $\tilde{\varphi}$ is similar to Theorems \[t43\] and \[t44\] and thus we skip it. Let $R$ be a $\kappa$-algebra. Based on the description of the morphism $\varphi_i$ explained in Section \[red\], $\mathrm{Ker~}\tilde{\varphi}(R)$ is the subgroup of $\tilde{M}(R)$ defined by the following conditions:

1.  If $i$ is even and $L_i$ is *of type I*, $s_i=\mathrm{id}$ mod $\pi \otimes 1$.

2.  If $i$ is even and $L_i$ is *of type II*, $m_{i,i}=\mathrm{id}$ mod $\pi \otimes 1$.

3.  Let $i$ be even and $L_i$ be *bound of type II*. Then $\delta_{i-1}^{\prime}e_{i-1}\cdot m_{i-1, i}+\delta_{i+1}^{\prime}e_{i+1}\cdot m_{i+1, i}+\delta_{i-2}e_{i-2}\cdot m_{i-2, i}+\delta_{i+2}e_{i+2}\cdot m_{i+2, i}=0$ mod $\pi \otimes 1$.\
    Let $i$ be even and $L_i$ be *of type I*. Then $v_i(\mathrm{resp.~}(y_i+\sqrt{\bar{\gamma}_i}v_i))+(\delta_{i-2}e_{i-2}\cdot m_{i-2, i}+\delta_{i+2}e_{i+2}\cdot m_{i+2, i})\tilde{e_i}=0$ mod $\pi \otimes 1$ if $L_i$ is *of type* $\textit{I}^o$ (resp. *of type* $\textit{I}^e$).

    Here,

    -   $
        \delta_{j}^{\prime} = \left\{
          \begin{array}{l l}
          1    & \quad  \textit{if $j$ is odd and $L_j$ is \textit{free of type I}};\\
          0    &   \quad  \textit{otherwise}.
            \end{array} \right.
        $

    -   If $j$ is odd, then $e_{j}=(0,\cdots, 0, 1)$ of size $1\times n_{j}$.

    -   When $j$ is even, $e_{j}=(0,\cdots, 0, 1)$ (resp. $e_j=(0,\cdots, 0, 1, 0)$) of size $1\times n_