Corollary 2

Tag 00X3, permanent. Cite it as [Bourbaki, Tag 00X3].

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

Let $\mathfrak{g}$ be a Lie subalgebra of $\mathfrak{g}\mathfrak{l}(V)$ . Then $[\mathfrak{g},\mathfrak{g}]$ is decomposable.

Let $\mathfrak{r}$ be the radical of $\mathfrak{g},\mathfrak{s}$ a Levi subalgebra of $\mathfrak{g}$ (Chap. I, §6, no. 8). Then

[\mathfrak{g},\mathfrak{g}] = [\mathfrak{s},\mathfrak{s}] + [\mathfrak{s},\mathfrak{r}] + [\mathfrak{r},\mathfrak{r}] =\mathfrak{s}+ [\mathfrak{g},\mathfrak{r}]

The algebra $[\mathfrak{g},\mathfrak{r}]$ is decomposable since all of its elements are nilpotent (Chap. I, §5, no. 3). On the other hand, $\mathfrak{s}$ is decomposable (Prop. 2). It follows that $[\mathfrak{g},\mathfrak{g}]$ is decomposable (Cor. 1).

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

Chap. I, §5, no. 3
Chap. I, §6, no. 8
Prop. 2
Cor. 1