Let $g$ be a Lie ring (Lie algebra over $\mathbb Z$), and let $U(g)$ and $S(g)$ denote the universal enveloping and symmetric algebra of $g$. The Poincaré-Birkhoff-Witt theorem (in the form proved by Lazard, see "Sur les algebres enveloppantes universelles de certaines algebres de Lie, M Lazard - Publ. Sci. Univ. Alger. Ser. A, 1954) yields a ring isomorphism between $S(g)$ and an associated graded of $U(g)$.

I can prove that $S(g)$ and $U(g)$ are isomorphic as $\mathbb Z$-modules; this essentially follows from the proof by Lazard. Is this already known?

isa ring isomorphism between $gr(U(g))$ and $S(g)$. Indeed $\mathbb Z$ is a PID. The typical "counterexamples" to PBW involve algebras over non-domains such as $\mathbb F_p[a,b,c]/(a^p,b^p,c^p)$. My question is about isomorphism of $S(g)$ and $U(g)$ as abelian groups, not as rings. – grok Aug 8 '12 at 11:29