In PGL$_{n}(\textbf{C})$, conjugacy classes of toral involutions can be represented by $$s_{i} = \begin{pmatrix}
-I_{i} & 0\\
0 & I_{n-i}
\end{pmatrix}.$$ for $1 \leqslant i \leqslant [n/2]$. Here an element is toral if it is contained in a conjugate of a fixed maximal torus of the algebraic group.

In $G =$ PGL$_{4}(\textbf{C})$, the toral elementary abelian 2-subgroups $E$ and their info interested are listed below. Denote $C_{G}(E)$ and $N_{G}(E)$ by $C$ and $N$, respectively. $T_{m}$ is a $m$-dimensional torus. 

|generators|cls dist| $C^{\circ}$ | $C/C^{\circ}$  |$N/C$|
|:----------:|:------:|:--------------------:|:------------:|:-------:|
|$\begin{pmatrix}
-1 & 0\\
0 & I_{3}
\end{pmatrix}$  |[1,0]|$A_{2}$| 1                           | 1|
|$\begin{pmatrix}
-I_{2} & 0\\
0 & I_{2}
\end{pmatrix}$  |[0,1]|$A_{1}A_{1}$|2                            |1|
|$\begin{pmatrix}
-1 & 0\\
0 & I_{3}
\end{pmatrix},\begin{pmatrix}
1 & 0 &0\\
0 & -1 &0 \\
0&0&I_{2}
\end{pmatrix}$  |[2,1]|$A_{1}$|1|2|
|$\begin{pmatrix}
-I_{2} & 0\\
0 & I_{2}
\end{pmatrix},\begin{pmatrix}
-1 & 0 &0 &0\\
0 & 1 &0 &0\\
0&0&-1&0\\
0&0&0&1
\end{pmatrix}$  |[0,3]|$T_{3}$|$V_{4}$|$S_{3}$|
|$\begin{pmatrix}
-1 & 0\\
0 & I_{3}
\end{pmatrix},\begin{pmatrix}
1 & 0 &0\\
0 & -1 &0 \\
0&0&I_{2}
\end{pmatrix},\begin{pmatrix}
1 & 0 &0 &0\\
0 & 1 &0 &0\\
0&0&-1&0\\
0&0&0&1
\end{pmatrix}$  |[4,3]|$T_{3}$|1|$S_{4}$|


To embed these five groups in $H$ = PGL$_{8}(\textbf{C})$, we replace each entry $a$ of the generators by $a I_{2}$. How the class distribution and the $C^{\circ}$ change seems obvious. But it is not quite clear to me that the structure of $C/C^{\circ}$ and $N/C$ is preserved by the embedding.

|cls dist| $C^{\circ}$ | $C/C^{\circ}$  |$N/C$|
|:------:|:--------------------:|:------------:|:-------:|
|[0,1,0,0]|$A_{1}A_{5}$| 1                           | 1|
|[0,0,0,1]|$A_{3}A_{3}$|2                            |1|
|[0,2,0,1]|$A_{1}A_{3}A_{1}$|1|2|
|[0,0,0,3]|$A_{1}A_{1}A_{1}A_{1}$|$V_{4}$|$S_{3}$|
|[0,4,0,3]|$A_{1}A_{1}A_{1}A_{1}$|1|$S_{4}$|

Question: Similar observation holds for toral elementary abelian 2-subgroups of PGL$_{3}(\textbf{C})$ embedded in PGL$_{6}(\textbf{C})$, of PGL$_{5}(\textbf{C})$ in PGL$_{10}(\textbf{C})$, etc. So does this embedding preserve the connectedness of the centraliser in general when we consider the toral elementary abelian 2-subgroups of $\operatorname{PGL}_{n}(\textbf{C})$ embedded in PGL$_{2n}(\textbf{C})$ ? Or better yet, does it preserve $C/C^{\circ}$ and $N/C$?

Edit: I just learned this is in fact a Kronecker product of the generators with the identity matrix $I_{2}$. So the question boils down to whether this product preserve the connectedness of the group centralizer...