component $X$ corresponding to the leftmost $0$ of the $1$-signature of $M$, we define $\tilde f_1 M$ as follows: $$\label{lem1}
          \tilde f_1 M=
          \begin{cases}
          MX_{\bar2}(-1)^{(0,-1)}X_{\bar1}(-1)^{(0,1)}=MA_1(1)^{-1}
                \quad\textup{if $X={X_{\bar2}(-1)}^{(0,b_{\bar2}^{-1})}$},\\
          MX_0(-1)^{(0,-1)}X_{\bar3}(-1)^{(0,1)}=MA_1(0)^{-1}
                \quad\textup{if $X={X_0(-1)}^{(0,b_0^{-1})}$},\\
          MX_3(-1)^{(0,-1)}X_0(-1)^{(0,1)}=MA_1(0)^{-1}
                \quad\textup{if $X={X_3(-1)}^{(0,b_3^{-1})}$}.
          \end{cases}$$ We define $$\label{lem3}
          \tilde f_1 M=M{X_1(-1)}^{(0,-1)}{X_{2}(-1)}^{(0,1)}
          =MA_1(-1)^{-1}$$ if no $0$ remains.

-   Depending on the component $X$ corresponding to the rightmost $1$ of the $1$-signature of $M$, we define $\tilde e_1 M$ as follows: $$\tilde e_1 M=
          \begin{cases}
          M X_{\bar2}(-1)^{(0,1)}X_{\bar1}(-1)^{(0,-1)}=M A_1(1)
                \quad \textup{if $X={X_{\bar 1}(-1)}^{(0,b_{\bar 1}^{-1})}$},\\
          M X_0(-1)^{(0,1)}X_{\bar3}(-1)^{(0,-1)}=M A_1(0)
                \quad \textup{if $X={X_{\bar3}(-1)}^{(0,b_{\bar3}^{-1})}$},\\
          M X_3(-1)^{(0,1)}X_0(-1)^{(0,-1)}=M A_1(0)
                \quad \textup{if $X={X_0(-1)}^{(0,b_0^{-1})}$},\\
          M {X_1(-1)}^{(0,1)}{X_{2}(-1)}^{(0,-1)}=M A_1(-1)
                \ \textup{if $X={X_2(-1)}^{(0,b_2^{-1})}$}.
          \end{cases}$$ We define $\tilde e_1 M=0$ if no $1$ remains.

Kashiwara actions $\tilde f_2$ and $\tilde e_2$ :

-   Consider the following finite ordered sequence of some components of $M$. $${X_{\bar2}(-1)}^{(0,b_{\bar2}^{-1})}
          {X_{\bar3}(-1)}^{(0,b_{\bar3}^{-1})}
          {X_3(-1)}^{(0,b_3^{-1})}
          {X_2(-1)}^{(0,b_2^{-1})}
          {X_3(-2)}^{(0,b_3^{-2})}.$$

-   Under each of the components $${X_{\bar2}(-1)}^{(0,b_{\bar2}^{-1})},
          {X_3(-1)}^{(0,b_3^{-1})},
          {X_3(-2)}^{(0,b_3^{-2})},$$ from the above sequence, write $b_j^k$-many $1$’s, and under each $${X_{\bar3}(-1)}^{(0,b_{\bar3}^{-1})},
          {X_2(-1)}^{(0,b_2^{-1})},$$ write $b_j^{-1}$-many $0$’s.

-   From this sequence of $1$’s and $0$’s, successively cancel out each $(0,1)$-pair to obtain a sequence of $1$’s followed by $0$’s. This remaining $1$ and $0$ sequence is called the .

-   Depending on the component $X$ corresponding to the leftmost $0$ 