For the algebraic group $PGL_4(K)$ where $K$ is algebraically closed with char not eq $2$, we have an elementary abelian $2-$subgroup
$$E = \left<\begin{pmatrix} -1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix},\begin{pmatrix} -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}\right>.$$
I can obtain myself $C_G(E)^{\circ} = T_3$ ($3$-dim torus) and I obtained the info that $C_G(E)/C_G(E)^{\circ} \cong V_4$ (the Klein $4$ group) by direct computing. What is the action of this $V_4$ on $T_3$? Do the non-identity elements act by inverting the three tori?