Proposition 6

Tag 0A33, permanent. Cite it as [Bourbaki, Tag 0A33].

Lie Groups and Lie Algebras, chapter II, § 2. Free Lie algebras

Let $g$ be a Lie algebra and $(\mathscr{C}^n g)_{n \geq 1}$ the lower central series of $g$ . Then

[\mathscr{C}^m g, \mathscr{C}^n g] \subset \mathscr{C}^{m+n} g \quad \text{for } m \geq 1 \text{ and } n \geq 1.

Generalizing Definition 1 of Chapter I, § 4, no. 1, we shall say that a Lie algebra $g$ is nilpotent if $\mathscr{C}^n g = \{0\}$ for $n$ sufficiently large. The nilpotency class of a nilpotent Lie algebra $g$ is the smallest integer $n$ such that $\mathscr{C}^{n+1} g = \{0\}$ .

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

Cited by

Nothing.

Cites

Definition 1 of Chapter I, § 4, no. 1