Corollary 1

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

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

Let g be a Lie algebra. Then g is solvable if and only if $\mathscr{D}g$ is orthogonal to g with respect to the Killing form.

This is an immediate consequence of Proposition 5 (b).

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

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