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