$a_{1}^{[i]}U_{1}^{\dagger }$ if $|i|=1$, $a_{0}^{[i]}$ for $a_{0}^{[i]}U_{1}^{\dagger }$, $a_{1}^{[i]}$ for $a_{1}^{[i]}U_{2}^{\dagger }$ if $|i|=2$, and $a_{0}^{[i]}$ for $a_{0}^{[i]}U_{2}^{\dagger }$ if $|i|=3$. It is further required that the dimensions of the four charge sectors are the same. Then, by permuting the order of basis, $(0,1,2,3)\rightarrow (0,2,1,3)$, these local matrices can display an even $\mathbb{Z}_{2}$ graded structure: $$\begin{aligned}
A^{[i]} &=&\left[ 
\begin{array}{cccc}
a_{0}^{[i]} & 0 & 0 & 0 \\ 
0 & a_{0}^{[i]} & 0 & 0 \\ 
0 & 0 & a_{1}^{[i]} & 0 \\ 
0 & 0 & 0 & a_{1}^{[i]}\end{array}\right] ,|i|=0;  \notag  \label{Z4matrices} \\
A^{[i]} &=&\left[ 
\begin{array}{cccc}
0 & 0 & a_{0}^{[i]} & 0 \\ 
0 & 0 & 0 & a_{0}^{[i]} \\ 
0 & a_{1}^{[i]} & 0 & 0 \\ 
a_{1}^{[i]} & 0 & 0 & 0\end{array}\right] ,|i|=1;  \notag \\
A^{[i]} &=&\left[ 
\begin{array}{cccc}
0 & a_{0}^{[i]} & 0 & 0 \\ 
a_{0}^{[i]} & 0 & 0 & 0 \\ 
0 & 0 & 0 & a_{1}^{[i]} \\ 
0 & 0 & a_{1}^{[i]} & 0\end{array}\right] ,|i|=2;  \notag \\
A^{i} &=&\left[ 
\begin{array}{cccc}
0 & 0 & 0 & a_{0}^{[i]} \\ 
0 & 0 & a_{0}^{[i]} & 0 \\ 
a_{1}^{[i]} & 0 & 0 & 0 \\ 
0 & a_{1}^{[i]} & 0 