& 0 & 0 \\ 
0 & \mathbbm{1} & 0 & 0 & U_{2} & 0 \\ 
0 & 0 & \mathbbm{1} & 0 & 0 & U_{3} \\ 
U_{1}^{\dagger } & 0 & 0 & \mathbbm{1} & 0 & 0 \\ 
0 & U_{2}^{\dagger } & 0 & 0 & \mathbbm{1} & 0 \\ 
0 & 0 & U_{3}^{\dagger } & 0 & 0 & \mathbbm{1}\end{array}\right] ,$$where $U_{1}$, $U_{2}$ and $U_{3}$ are unitary matrices required by the idemponency. Applying $P_{0}$ to $A^{[i]}$, we obtain $$A^{[i]}=\text{diag}\left(
a_{0}^{[i]},a_{1}^{[i]},a_{2}^{[i]},a_{0}^{[i]},a_{1}^{[i]},a_{2}^{[i]}\right) \times \left( Y_{6}^{|i|}\otimes \mathbbm{1}\right)$$with some redefinitions. Via permuting the basis $(0,1,2,3,4,5,6)\rightarrow
(0,2,4,1,3,5)$, we can rewrite $A^{[i]}$ into the standard form on the $\mathbb{Z}_{2}$ parity basis: $$\begin{aligned}
A^{[i]} &=&\left[ 
\begin{array}{cc}
d^{[i]} & 0 \\ 
0 & U^{\dagger }d^{[i]}U\end{array}\right] ,|i|=0,2,4;  \notag \\
A^{[i]} &=&\left[ 
\begin{array}{cc}
0 & d^{[i]} \\ 
U^{\dagger }d^{[i]}U^{\dagger } & 0\end{array}\right] ,|i|=1,3,5,\end{aligned}$$where $$\begin{aligned}
U &=&\left[ 
\begin{array}{ccc}
0 & \mathbbm{1} & 0 \\ 
0 & 0 & \mathbbm{1} \\ 
\mathbbm{1} & 0 & 0\end{array}\right] ,  \notag \\
d^{[i]} &=&\left[ 
\begin{array}{ccc}
a_{0}^{[i]} & 0 & 0 \\ 
0 & a_{2}^{[i]} & 0 \\ 
0 & 0 & a_{1}^{[i]}\end{array}\right] ,|i|=0,1;  \notag \\
d^{[i]} &=&\left[ 
\begin{array}{ccc}
0 & a_{0}^{[i]} & 0 \\ 
0 & 0 & 