Let $e_{1}$ and $e_{2}$ be involutions in the algebraic group $G=\operatorname{PGL}_{n}(\mathbb{C})$. Do we have $$C_{G}(\langle e_{1},e_{2}\rangle)^{\circ} = C_{G}(e_{1})^{\circ}\cap C_{G}(e_{2})^{\circ}.$$ It's obvious $C_{G}(\langle e_{1},e_{2}\rangle)^{\circ} \leqslant C_{G}(e_{1})^{\circ}\cap C_{G}(e_{2})^{\circ}.$ If yes, can it be generalised to two elementary abelian $2$-subgroups $E_{1}, E_{2}$.
Here $H^{\circ}$ denotes the identity component of the group $H$.