Corollary 2

Tag 00WT, permanent. Cite it as [Bourbaki, Tag 00WT].

Lie Groups and Lie Algebras, chapter VII, § 5. Decomposable linear Lie algebras

Let $\mathfrak{r}$ be the radical of $\mathfrak{g}$ . If $\mathfrak{g}$ is decomposable, $\mathfrak{r}$ is decomposable.

Indeed, $e(\mathfrak{r})$ is a solvable ideal of $\mathfrak{g}$ by Prop. 4 and Cor. 1, hence $e(\mathfrak{r}) =\mathfrak{r}$ .

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

Proposition 6, § 5
Proposition 7, § 5
Theorem 2, § 5

Cites

Prop. 4
Cor. 1