Proposition 4

Tag 017E, permanent. Cite it as [Bourbaki, Tag 017E].

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

Let $h$ be a non-zero element of $\mathfrak{g}$ . Then $h$ is simple if and only if there exists $x\in \mathfrak{g}$ such that $[h, x] = 2x$ and $h\in$ (ad $x$ ) $(\mathfrak{g})$ .

The condition is clearly necessary. It is sufficient by Lemma 6.

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

Nothing.

Cites

Lemma 6