In the algebraic group $G = \operatorname{PGL}_4(\mathbb{C})$, let $E$ denote the subgroup of elements of order dividing 2 in the diagonal maximal torus; it is generated by the images of the three matrices $$ \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{bmatrix},\quad \begin{bmatrix} 1 & 0 & 0\\ 0 & -1 & 0 \\ 0 & 0 & I_2 \end{bmatrix},\quad \begin{bmatrix} -1 & 0 \\ 0 & I_3 \end{bmatrix}. $$ Direct computation shows that $N_{G}(E)/C_{G}(E) \cong S_4\,$, which is the Weyl group of $G$. Analogous result can be obtained for $n = 3, 5, 6, 7, 8, 9, 10$ in $\operatorname{PGL}_{n}(\mathbb{C})$ by direct calculations. Is this observation true for any $n$? Is there any reference relevant?