Proposition 1

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

Lie Groups and Lie Algebras, chapter VII, § 2. Cartan subalgebras and regular elements of a Lie algebra

Let $\mathfrak{g}$ be a Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$ . Then $\mathfrak{h}$ is a maximal nilpotent subalgebra of $\mathfrak{g}$ .

Let $\mathfrak{h}'$ be a nilpotent subalgebra of $\mathfrak{g}$ containing $\mathfrak{h}$ . Then $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{h}'$ (Example 3), so $\mathfrak{h}=\mathfrak{h}'$ (Example 1).

There exist maximal nilpotent subalgebras that are not Cartan subalgebras (Exerc. 2).

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 2, § 2

Cites

Exerc. 2
Example 1
Example 3