Example 4

\* Let $K$ be a commutative ring, $\mathfrak g$ a Lie algebra over $K$ and $U$ the enveloping algebra of $\mathfrak g$. An increasing filtration $(U_n)_{n\in\mathbf{Z}}$ is defined on $U$ by taking $U_n=\{0\}$ for $n<0$ and denoting by $U_n$, for $n\geq0$ the set of elements of $U$ which can be expressed as a sum of products of at most $n$ elements of $\mathfrak g$, then

U_0 = K and gr(U) is a commutative K-algebra (Lie Groups and Lie Algebras, Chapter I, § 2, no. 6). The canonical mapping of g to gr_1(U) = U_1/U_0 can be extended uniquely to a homomorphism h of the symmetric algebra $S(g)$ of the K-module g to the K-algebra gr(U); the homomorphism h is surjective and, if the K-module g is free, h is bijective (loc. cit., no. 7, Theorem 1).