n{array}{l l}
    v_i\cdot ({}^tg_{i, i}-\mathrm{Id}_{n_i})   &   \quad  \textit{if $L_i$ is \textit{free of type I}};\\
      \delta_{i-1}v_{i-1}\cdot {}^tg_{i, i-1}+\delta_{i+1}v_{i+1}\cdot {}^tg_{i, i+1}   & \quad  \textit{if $L_i$ is \textit{bound of type I}}.
        \end{array} \right.$$

    Here,

    -   $v_{i}=(0,\cdots, 0, 1, 0)$ of size $1\times n_{i}$ and $\mathrm{Id}_{n_i}$ is the identity matrix of size $n_i \times n_i$.

    -   $v_{i-1}$ (resp. $v_{i+1}$)$=(0,\cdots, 0, 1)$ of size $1\times n_{i-1}$ (resp. $1\times n_{i+1}$).\

    Then each entry of the above matrix lies in the ideal $(\pi)$. If $L_i$ is *of type II*, then $A_i=B_i$ and $B_i^{\perp}=A_i^{\perp}$ so that there is no contribution.\

Construction of ** {#m}
------------------

We define a functor from the category of commutative flat $A$-algebras to the category of monoids as follows. For any commutative flat $A$-algebra $R$, let $$\underline{M}(R) \subset \{m \in \mathrm{End}_{B\otimes_AR}(L \otimes_A R)\}$$ to be the set of $m \in \mathrm{End}_{B\otimes_AR}(L \otimes_A R)$ satisfying the following conditions:

-   $m$ stabilizes $A_i\otimes_A R,B_i\otimes_A R,W_i\otimes_A R,X_i\otimes_A R$ for all $i$ and $Z_i\otimes_A R$ for all even integer $i$.

-   $m$ induces the identity on $A_i\otimes_A R/ B_i\otimes_A R$ for all $i$.

-   $m$ induces the identity on $W_i\otimes_A R/(X_i\cap Z_i)\otimes_A R$ for all even integer $i$.

-   $m$ induces the identity on $B_i^{\perp}\otimes_A R/ A_i^{\perp}\otimes_A R$ for all odd integer $i$.

\[r31\] We give another description for the functor $\underline{M}$. Let us define a functor from the category of commutative flat $A$-algebras to the category of rings as follows:

For any commutative flat $A$-algebra $R$, define $$\underline{M}^{\prime}(R) \subset \{m \in \mathrm{End}_{B\otimes_AR}(L \otimes_A R) \}$$ to be the set of $m\in  \mathrm{End}_{B\otimes_AR}(L \otimes_A R)$ satisfying the following conditions:

-   $m$ stabilizes $A_i\otimes_A R,B_i\otimes_A R,W_i\otimes_A R,X_i\otimes_A R$ f