are still given by $P_{i}=Q_{4}^{i}P_{0}Q_{4}^{-i}$ with $i=0,1,2,3 $, where each of $P_{i}$ is associated with an irreducible invariant subspace. The discussion is divided into three different situations.

1\. $\left[ P_{0},Q_{4}\right] =0$

Since the irreducible invariant subspace projector $P_{0}$ contains four different charge sectors, the forms of the local matrices $A^{[i]}$ are given by Eq. (\[block\]), and the irreducible $A^{[i]}$ span a trivial simple $\mathbb{Z}_{4}$ graded algebra. Both the transfer matrix spectrum and the entanglement spectrum are not necessarily degenerate.

2\. $\left[ P_{0},Q_{4}\right] \neq 0$ but $\left[ P_{0},Q_{4}^{2}\right] =0$

In this case we have only two orthogonal projectors $P_{0}$ and $P_{1}$ for two irreducible invariant subspaces. According to $Q_{4}P_{0}Q_{4}^{-1}+P_{0}=$ $\mathbbm{1}$ and $P_{0}^{2}=P_{0}$, $P_{0}$ can be determined as $$P_{0}=\frac{1}{2}\left[ 
\begin{array}{cccc}
\mathbbm{1} & 0 & U_{1} & 0 \\ 
0 & \mathbbm{1} & 0 & U_{2} \\ 
U_{1}^{\dagger } & 0 & \mathbbm{1} & 0 \\ 
0 & U_{2}^{\dagger } & 0 & \mathbbm{1}\end{array}\right] ,$$where $U_{1}$ and $U_{2}$ are unitary block matrices and both the charge-0 and charge-1 sectors have the same dimension as the charge-2 and charge-3 sectors. The relation $A^{[i]}P_{j}=P_{j}A^{[i]}P_{j}$ and the gauge transformation $G=\mathbbm{1}\oplus \mathbbm{1}\oplus U_{1}\oplus U_{2}$ lead to $$A^{[i]}=\text{diag}\left(
a_{0}^{[i]},a_{1}^{[i]},a_{0}^{[i]},a_{1}^{[i]}\right) \times \left(
Y_{4}^{|i|}\otimes \mathbbm{1}\right)$$where we have substituted $a_{1}^{[i]}$ for 