Centralizer $C_G(T)$ of a maximal torus $T$ is in every Borel subgroup containing $T$?

Question

I read that if $T$ is a maximal torus of a connected algebraic group $G$, then $C_G(T)$ is in every Borel subgroup containing $T$.

I know $C_G(T)=N_G(T)^\circ$, the connected component of the identity in $N_G(T)$, hence is connected. Since $T$ is maximal, $C_G(T)$ is nilpotent, hence solvable, thus contained in some Borel group $B$. Then $T\subset C_G(T)\subset B$, so I get that $C_G(T)$ is in at least one Borel subgroup containing $T$.

How does it follow that $C_G(T)$ is in all of them? I know that Borel subgroups are conjugate, But if $T\subset gBg^{-1}$ for some $g\in G$, I can't see why $C_G(T)\subset gBg^{-1}$ still.

Answer 1

Here's a hint: maximal tori of $gBg^{-1}$ are conjugate.