Lemma 1

The ideal $a$ of Definition 1 is graded. For $a, b$ in Lib(X), let $B(a, b) = a.b + b.a$. The formulae

for $w_1, \ldots, w_n$ in $M(X)$ and $\lambda_1, \ldots, \lambda_n$ in $K$, show that the families $(Q(a))_{a \in \mathrm{Lib}(X)}$ and $(Q(w), B(w, w'))_{w, w' \in M(X)}$ generate the same submodule of $\mathrm{Lib}(X)$. As J is trilinear the ideal $a$ is generated by the homogeneous elements $Q(w), B(w, w')$ and $J(w, w', w'')$, where $w, w', w''$ are in $M(X)$, and hence is graded (Algebra, Chapter III, § 3, no. 3, Proposition 1).

Let the Lie algebra $L(X) = \mathrm{Lib}(X)/a$ be given the quotient graduation. The homogeneous component of $L(X)$ of degree $\delta$ is denoted by $L^\delta(X)$; it is the submodule of $L(X)$ generated by the images of the elements $w \in M(X)$ such that $\phi(w) = \delta$.

We shall make special use of the following two cases: (a) Total graduation: we take $\Delta = \mathbf{N}$ and $\phi_0(x) = 1$ for all $x \in X$, whence $\phi(w) = l(w)$ for $w$ in $M(X)$. The K-module $L^n(X)$ is generated by the images of the elements of length $n$ in $M(X)$, which we shall call alternants of degree n. We shall see later that the module $L^n(X)$ is free and admits a basis consisting of alternants of degree $n$ (no. 11, Theorem 1). Then $L(X) = \bigoplus_{n \geq 1} L^n(X)$ and $L^1(X)$ admits $X$ as basis (no. 2, Corollary 1 to Proposition 1). By the construction of $M(X)$,

and in particular

(b) Multigraduation: we take $\Delta$ to be the free commutative monoid $\mathbf{N}^{(X)}$ constructed on $X$. The mapping $\phi_0$ of $X$ into $\Delta$ is defined by $(\phi_0(x))(x') = \delta_{xx'}$, where $\delta_{xx'}$ is the Kronecker symbol. For $w \in M(X)$ and $x \in X$, the integer $(\phi(w))(x)$ is "the number of occurrences of the letter $x$ in $w$". For $\alpha$ in $\mathbf{N}^{(X)}$, we write $|\alpha| = \sum_{x \in X} \alpha(x)$, whence $|\phi(w)| = l(w)$ for all $w$ in $M(X)$. It follows that

obviously