Theorem 1

Let g be a Lie K-algebra, U its enveloping algebra, G the graded algebra associated with the filtered algebra U and S the symmetric algebra of the K-module g. If g is a free K-module, the canonical homomorphism $\omega : S \to G$ is an isomorphism.

Let $(x_\lambda)_{\lambda \in \Lambda}$ be a basis of the K-module g; we give $\Lambda$ a total ordering (Set Theory, Chapter III, § 2, no. 3, Theorem 1). Let P be the polynomial algebra $K[z_\lambda]_{\lambda \in \Lambda}$ in indeterminates $z_\lambda$ in one-to-one correspondence with the $x_\lambda$. For every sequence $M = (\lambda_1, \lambda_2, ..., \lambda_n)$ of elements of $\Lambda$, let $z_M$ denote the monomial $z_{\lambda_1}z_{\lambda_2}...z_{\lambda_n}$ and $x_M$ the tensor $x_{\lambda_1} \otimes x_{\lambda_2} \otimes \cdots \otimes x_{\lambda_n}$.

The $z_M$, for M increasing, form a basis of the K-module P (we make the convention that $\varnothing$ is an increasing sequence and that $z_\varnothing = 1$). Let $P_p$ be the submodule of polynomials of degree $\leq p$. We shall first prove several lemmas. (To abbreviate, we write $\lambda \leq M$ if $\lambda \leq \mu$ for every index $\mu$ of the sequence M.)