Proposition 3

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

Lie Groups and Lie Algebras, chapter I, § 4. Nilpotent Lie algebras

Let $g$ be a nilpotent Lie algebra and $\mathfrak{h}$ a subalgebra of $g$ distinct from $g$ . The normalizer of $\mathfrak{h}$ in $g$ is distinct from $\mathfrak{h}$ . Let $k$ be the greatest integer such that $\mathscr{C}^k g + \mathfrak{h} \neq \mathfrak{h}$ . Then

[\mathscr{C}^k g + \mathfrak{h}, \mathfrak{h}] \subset \mathscr{C}^{k+1} g + \mathfrak{h} \subset \mathfrak{h}

and hence the normalizer of

\mathfrak{h}

g

contains

\mathscr{C}^k g + \mathfrak{h}

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

Cited by

§ 4. Nilpotent Lie algebras
Exercise 4, § 4
Example 1, § 2

Cites

Nothing.