Proposition 4

Tag 0A4Q, permanent. Cite it as [Bourbaki, Tag 0A4Q].

Lie Groups and Lie Algebras, chapter II, § 4. Central filtrations

(i) The lower central series of G is an integral central filtration on G.

(ii) If (G_n)_{n \in \mathbf{N}^*} is an integral central filtration on G, then C^nG \subset G_n for all n \in \mathbf{N}^*.

Assertion (i) has been proved in Algebra, Chapter I, § 6, no. 3, formula (7).

We prove (ii) by induction on n; C^1G = G = G_1; for n > 1,

C^nG = (G, C^{n-1}G) \subset (G, G_{n-1}) \subset G_n.

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

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