Isomorphism between reductive groups implies isomorphic Levi subgroups of prescribed type?

Question

I've come up with the following question.

Assume that $G$ and $G'$ are two isomorphic reductive algebraic groups over an algebraically closed field $k$. If $P$ (resp. $P'$) is the standard parabolic subgroup of $G$ of type $\Theta\subset Dyn(G)$ (resp. of $G'$ of type $\Theta\subset Dyn(G')$), are the respective Levi subgroups $L_{P}$ and $L_{P'}$ of $P$ and $P'$ isomorphic? I think it should be the case but i didn't figure out a proof.

edit : here is what i tried : if $f$ is the isomorphism between $G$ and $G$. The Levi subgroup associated to the standart parabolic subgroup of type $\Theta$ in $G$ is the centralizer of $C_{G'}(T_{\Theta})$ of $T_{\Theta}$, where $T_{\Theta}=\left(\bigcap_{\alpha\in \Theta} \ker(\alpha)\right)$ ($\alpha$ are roots in $G$). Since we have $f(C_G(T_{\Theta}))=C_{G'}(f(T_{\Theta}))$, it would suffice to show that $f(T_{\Theta})=\left(\bigcap_{\beta\in \Theta} \ker(\beta)\right)$ where the $\beta$ are the roots in $G$...

Answer 1

You are writing $\Theta$ to denote a subset of the roots of both $G$ and $G'$; how are you identifying these two subsets? If the identification is via the isomorphism $f$, then the answer to your question is tautologically yes. If you have some other identification in mind, you should explain what it is.