Corollary 1

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

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

If $\mathfrak{a}$ is a decomposable Lie subalgebra of $\mathfrak{g}\mathfrak{l}(V)$ , and $\mathfrak{n}$ a Lie subalgebra of $\mathfrak{a}$ , the normalizer (resp. centralizer) of $\mathfrak{n}$ in $\mathfrak{a}$ is decomposable.

This follows from Prop. 3 by taking $\mathfrak{c}=\mathfrak{n},\mathfrak{b}=\mathfrak{n}$ (resp. $\mathfrak{c}=\mathfrak{n},\mathfrak{b}=\{0\}$ ).

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

Corollary 2, § 5
Theorem 2, § 10

Cites

Prop. 3