This may be a dumb question.
I want to prove a following lemma.
Let $V,V'$ be vector spaces of dimension $r$ over $\mathbb{C}$.
Assume there is an isomorphism $\Phi$ between the flag varieties, $\Phi:\operatorname{Fl}(V)\rightarrow \operatorname{Fl}(V')$.
Then, $\Phi$ induces an isomorphism as Lie algera, $\Psi: \operatorname{End}_{0}(V)\rightarrow \operatorname{End}_{0}(V')$.
( $\operatorname{End}_{0}(V)$ is the Lie algebra consisting of trace-zero endomorphisms. )
I don't know how to define $\Psi$ explicitly and why it is an isomorphism as Lie algebra.
Thanks in advance.