y}{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) if $i$ is even and $L_i$ is *of type $I^o$* (resp. if $i$ is even and $L_i$ is *of type $I^e$*, or if $i$ is odd and $L_i$ is *free of type $I$*) and $y_i, v_i, z_i, r_i, t_i, y_i, x_i, u_i, w_i$ are matrices of suitable sizes.

3.  Assume that $i$ is even and that $L_i$ is *of type I*. Then $$z_i+\delta_{i-2}k_{i-2, i}+\delta_{i+2}k_{i+2, i}=\pi z_i^{\ast}$$ such that $z_i^{\ast}\in B\otimes_AR$. Here,

    -   $z_i$ is an entry of $m_{i,i}$ as described in the above step (b).

    -   $k_{i-2, i}$ (resp. $k_{i+2, i}$) is the $(n_{i-2}, n_i)^{th}$-entry (resp. $(n_{i+2}, n_i)^{th}$-entry) of the matrix $m_{i-2, i}$ (resp. $m_{i+2, i}$) if $L_{i-2}$ (resp. $L_{i+2}$) is *of type* $\textit{I}^o$.

    -   $k_{i-2, i}$ (resp. $k_{i+2, i}$) is the $(n_{i-2}-1, n_i)^{th}$-entry (resp. $(n_{i+2}-1, n_i)^{th}$-entry) of the matrix $m_{i-2, i}$ (resp. $m_{i+2, i}$) if $L_{i-2}$ (resp. $L_{i+2}$) is *of type* $\textit{I}^e$.

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

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

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