How to use the Invariance Lemma to prove that the diagonal subalgebra of classical Lie algebras are self-normalizing? When $\operatorname{char}\ F=0$?
(Invariance Lemma) Assume that $F$ has characteristic zero. Let $L$ be a Lie subalgebra of $gl(V )$ and let $A$ be an ideal of $L$. Let $λ : A → F$ be a weight of $A$. The associated weight space $V_λ = \{v ∈ V : av = λ(a)v,\ \forall a ∈ A\}$ is an L-invariant subspace of $V$.