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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 $V$.

Ronald
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    What is the "Invariance Lemma" ? I just would use that every CSA is self-normalizing, and these diagonal subalgebras of classical Lie algebras are just CSA's. – Dietrich Burde Feb 19 '16 at 09:44
  • Yes, I agree. But actually this question is to see an application of the Invariance Lemma which I will insert it under the question – Ronald Feb 19 '16 at 16:45
  • @DietrichBurde Sorry, but what do you mean by CSA? I’m quite puzzled with this abbreviation. (I have just covered the first two chapters of Humphrey’s book.) Thank you! – Hetong Xu Sep 16 '21 at 10:38
  • Usually a CSA is a Cartan subalgebra. Humphrey's says this in his book, I think. – Dietrich Burde Sep 16 '21 at 11:56

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