Corollary

Let $g$ be a Lie algebra and $(x_i)_{i \in I}$ a generating family of $g$. The $n$-th term $\mathscr{C}^n g$ of the lower central series of $g$ is the module generated by the iterated brackets $[x_{i_1}, [x_{i_2}, \ldots, [x_{i_{p-1}}, x_{i_p}] \ldots]]$ for $p \geq n$, and $i_1, \ldots, i_p$ in $I$. Let $f$ be the homomorphism of $L(I)$ into $g$ such that $f(i) = x_i$ for all $i \in I$. As $(x_i)_{i \in I}$ generates $g$, $g = f(L(I))$, whence $\mathscr{C}^n g = f(\mathscr{C}^n(L(I)))$ by Proposition 4 of Chapter I, § 1, no. 5. The corollary then follows from assertions (b) and (c) of Proposition 7.