# Inscrutable proof in Humphrey's book on Lie algebras and representations

This is a question pertaining to Humphrey's Introduction to Lie Algebras and Representation Theory

Is there an explanation of the lemma in §4.3-Cartan's Criterion? I understand the proof given there but I fail to understand how anybody could have ever devised it or had the guts to prove such a strange statement...

Lemma: let $k$ be alg. closed of caracteristic $0$, $V$ finite dimensional over $k$, $A\subset B\subset \mathrm{End}(V)$ subspaces and $M$ the set of endomorphisms $x$ of $V$ such that $[x,B]\subset A$. Suppose $x\in M$ is such that $\forall y\in M, \mathrm{Tr}(xy)=0$, then $x$ is nilpotent.

The proof uses the diagonalisable$+$nilpotent decomposition, and goes on to show that all eigenvalues of $x$ are $=0$ by showing that the $\mathbb{Q}$ subspace of $k$ they generate has only the $0$ linear functional.

Added: (t.b.) here's the page from Google books for those without access:

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It would help enormously if you reproduced at least the lemma, even better the proof, so that this question isn't only answerable by people with access to Humphreys. –  Qiaochu Yuan Apr 29 '12 at 9:38
Ok ^^ I was hoping to avoid it as I am wriring this from a phone but you're right. –  Olivier Bégassat Apr 29 '12 at 9:41
I included links and an image of the lemma and its proof. I hope that's what you intended. –  t.b. Apr 29 '12 at 9:51
@t.b. That would be lovely, do I have to do something for these to appear in the question? –  Olivier Bégassat Apr 29 '12 at 9:56
Your edit and mine were more or less simultaneous. I merged the two edits. Reload the page and see if that's okay now. –  t.b. Apr 29 '12 at 9:57