Lemma 3

Tag 0173, permanent. Cite it as [Bourbaki, Tag 0173].

Lie Groups and Lie Algebras, chapter VIII, § 11. Classes of nilpotent elements and sl2 -triplets

Let $x\in \mathfrak{g},\mathfrak{p}=$ Ker(ad $x$ ) $,\mathfrak{q}=$ Im(ad $x$ ). Then $[\mathfrak{p},\mathfrak{q}]\subset \mathfrak{q}$ , and $\mathfrak{p}\cap \mathfrak{q}$ is a subalgebra of $\mathfrak{g}$ .

If $u\in \mathfrak{p}$ and $v\in \mathfrak{q}$ , there exists $w\in \mathfrak{g}$ such that $v= [x, w]$ , so

[u, v] = [u,[x, w]] = [x,[u, w]]-[[x, u], w] = [x,[u, w]]\in \mathfrak{q}

On the other hand, $\mathfrak{p}$ is a subalgebra of $\mathfrak{g}$ , so $[\mathfrak{p}\cap \mathfrak{q},\mathfrak{p}\cap \mathfrak{q}]\subset \mathfrak{p}\cap \mathfrak{q}$ .

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

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Cites

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