Corollary 2

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

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

The radical r of a Lie algebra g is a characteristic ideal.

$\mathscr{D}g$ is a characteristic ideal and the Killing form is completely invariant (§ 3, no. 6, Proposition 10). Hence the orthogonal of $\mathscr{D}g$ with respect to the Killing form is a characteristic ideal (§ 3, no. 6, Proposition 7 (b)).

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

Cited by

Exercise 9, § 5
Corollary 3, § 5

Cites

§ 3, no. 6, Proposition 10
§ 3, no. 6, Proposition 7