Proposition 7

Let $X$ be a set and $n$ an integer $\geq 1$. (a) $L^{n+1}(X) = [L^1(X), L^n(X)]$. (b) The module $L^n(X)$ is generated by the elements $[x_1, [x_2, \ldots, [x_{n-1}, x_n] \ldots]]$ where $(x_1, \ldots, x_n)$ runs through the set of sequences of $n$ elements of $X$. (c) The lower central series of $L(X)$ is given by $\mathscr{C}^n(L(X)) = \sum_{p \geq n} L^p(X)$.

(a) We apply Proposition 5 with $g = L(X)$ and $P = L^1(X)$. By induction on $n$, we deduce from (12) (no. 6) and (19) the equality $P_n = L^n(X)$. The desired relation is then equivalent to the definition $[P, P_n] = P_{n+1}$. (b) This follows from (a) by induction on $n$. (c) Let $g = L(X)$ and $g_n = \sum_{p \geq n} L_p(X)$. Then $g = g_1$ and formula (13) of no. 6 implies $[g_n, g_m] \subset g_{n+m}$ and in particular $[g, g_n] \subset g_{n+1}$. By induction on $n$, $\mathscr{C}^n g \subset g_n$. On the other hand, from (a) we deduce that $L^n(X) \subset \mathscr{C}^n g$ by induction on $n$. As $\mathscr{C}^n g$ is an ideal of $g$, the relation $L^p(X) \subset \mathscr{C}^n g$ implies that

by (a). Hence $L^p(X) \subset \mathscr{C}^n g$ for $p \geq n$, whence $g_n \subset \mathscr{C}^n g$.