In the algebraic group $G = PGL_4(\mathbb{C})$, let $E$ be the unique (up to conjugacy) elementary abelian $2$-subgroup of maximal rank generated by the following three matrices: $ \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{bmatrix}, \begin{bmatrix} 1 & 0 & 0\\ 0 & -1 & 0 \\ 0 & 0 & I_2 \end{bmatrix},\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 $PGL_{n}(\mathbb{C})$ by direct calculation. Is this observation true for any $n$? Is there any reference relevant? Thank you!