Proposition 1

Let $\psi$ be the canonical mapping of Lib(X) onto $L(X)$ and $\phi$ the restriction of $\psi$ to X. For every mapping f of X into a Lie algebra $g$, there exists one and only one homomorphism F: $L(X) \to g$ such that $f = F \circ \phi$.

(a) Existence of F: let h be the homomorphism of Lib(X) into $g$ extending f (no. 1). For all $a$ in Lib(X), $h(Q(a)) = h(a.a) = [h(a), h(a)] = 0$; similarly, the Jacobi identity satisfied by $g$ implies that $h(J(a, b, c)) = 0$ for $a, b, c$ in Lib(X). It follows that $h(a) = 0$, whence there is a homomorphism F of $L(X)$ into $g$ such that $h = F \circ \psi$. By restricting to X, we obtain $f = F \circ \phi$.

(b) Uniqueness of F: let $F': L(X) \to g$ be a homomorphism such that $f = F' \circ \phi$. The homomorphisms $F \circ \psi$ and $F' \circ \psi$ of Lib(X) into $g$ coincide on X and hence are equal; as $\psi$ is surjective, $F = F'$.