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?