Proposition 5

Let $g$ be a Lie algebra and $P$ a submodule of $g$. We define the submodules $P_n$ of $g$ by the formulae $P_1 = P$ and $P_{n+1} = [P, P_n]$ for $n \geq 1$. Then

We prove (18) by induction on $m$. The case $m = 1$ is obvious. By the Jacobi identity, that is The induction hypothesis implies that $[P_m, P_{n+1}] \subset P_{m+n+1}$ and $[P_m, P_n] \subset P_{m+n}$, whence By formula (18), $P_n \supset \sum_{p=1}^{n-1} [P_p, P_{n-p}] \supset [P_1, P_{n-1}] = P_n$, whence (19).

When we take $P = g$, the sequence $(P_n)$ is the lower central series ($\mathscr{C}^n g$) of $g$ (Chapter I, § 1, no. 5). Hence: