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...
