Corollary 5

Tag 09YG, permanent. Cite it as [Bourbaki, Tag 09YG].

Lie Groups and Lie Algebras, chapter I, § 5. Solvable Lie algebras

For a Lie algebra $g$ to be solvable, it is necessary and sufficient that $\mathscr{D}g$ be nilpotent.

The condition is necessary by Corollary 1. It is sufficient since $g/\mathscr{D}g$ is commutative.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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