$v_{i+1}=(0,\cdots, 0, 1, 0)$) of size $1\times n_{i+1}$ if $L_{i+1}$ is *of type* $\textit{I}^o$ (resp. *of type* $\textit{I}^e$).

5.  Assume that $i$ is odd and that $L_i$ is *bound of type I*. Then $$\delta_{i-1}v_{i-1}\cdot {}^tm_{i, i-1}+\delta_{i+1}v_{i+1}\cdot {}^tm_{i, i+1} = \pi m_{i,i}^{\ast\ast}$$ such that $ m_{i,i}^{\ast\ast} \in M_{1\times n_i}(B\otimes_AR)$. Here, $v_{i-1}$ (resp. $v_{i+1}$)$=(0,\cdots, 0, 1)$ of size $1\times n_{i-1}$ (resp. $1\times n_{i+1}$).\

In conclusion, each element of $\underline{M}(R)$ for a flat $A$-algebra $R$ is written as the following matrix $$m= \begin{pmatrix} \pi^{max\{0,j-i\}}m_{i,j} \end{pmatrix} \mathrm{~together~with~}z_i^{\ast}, m_{i,i}^{\ast}, m_{i,i}^{\ast\ast}.$$

**\
However, for a general $R$, the above description for $\underline{M}(R)$ will no longer be true. For such $R$, we use our chosen basis of $L$ to write each element of $\underline{M}(R)$ formally. We describe each element of $\underline{M}(R)$ as formal matrices $$m= \begin{pmatrix} \pi^{max\{0,j-i\}}m_{i,j} \end{pmatrix} \mathrm{~with~}z_i^{\ast}, m_{i,i}^{\ast}, m_{i,i}^{\ast\ast}$$ such that

1.  When $i\neq j$, $m_{i,j}$ is an $(n_i \times n_j)$-matrix with entries in $B\otimes_AR$.

2.  When $i=j$, $m_{i,i}$ is an $(n_i \times n_i)$-formal matrix such that $$m_{i,i}=\left\{
      \begin{array}{l l}
      \begin{pmatrix} s_i&\pi y_i\\ \pi v_i&1+\pi z_i \end{pmatrix}    & \quad  \textit{if $i$ is even and $L_i$ is \textit{of type $I^o$}};\\
      \begin{pmatrix} s_i&r_i&\pi t_i\\ \pi y_i&1+\pi x_i&\pi z_i\\ v_i&u_i&1+\pi w_i \end{pmatrix}    & \quad  \textit{if $i$ is even and $L_i$ is \textit{of type $I^e$}};\\
    \begin{pmatrix} s_i&\pi r_i&t_i\\  y_i&1+\pi x_i& u_i\\\pi  v_i&\pi z_i&1+\pi w_i \end{pmatrix}
    & \quad  \textit{if $i$ is odd and $L_i$ is \textit{free of type $I$}};\\
      m_{i,i}   & \quad  \textit{otherwise}.
          \end{array} \right.$$ Here, $s_i$ is an $(n_i-1 \times n_i-1)$-matrix (resp. $(n_i-2 \times n_i-2)$-matrix) with entries in $B\otimes_AR$ if $i$ is ev