Theorem 1

Let $H$ be a Hall set relative to $X$ and $\Psi$ the canonical mapping of $M(X)$ into the free Lie algebra $L(X)$. The restriction of $\Psi$ to $H$ is a basis of the module $L(X)$.

For every element $w$ of $H$ we write $\bar{w} = \Psi(w)$.

(A) The case where $X$ is finite.

If $X$ is empty, so are $M(X)$ and therefore $H$ and $L(X)$ is zero. If $X$ has a single element $x$, $H \cap M^n(X)$ is empty for $n \geq 2$ (Proposition 12 (a)). Therefore, $H = \{x\}$; we know also (no. 2, Remark) that the module $L(X)$ is free and has basis $\{\bar{x}\}$. The theorem is therefore true when $X$ has at most one element.

Suppose henceforth that $X$ has at least two elements; choose sequences $(w_p)$ and $(P_p)$ with the properties stated in Proposition 13. For every integer $p \geq 0$, let $L_p$ denote the submodule of $L(X)$ generated by the elements $\bar{w}_i$ with $0 \leq i < p$ and $g_p$ the Lie subalgebra of $L(X)$ generated by the family $(\bar{u})_{u \in P_p}$.