The enveloping algebra is free over its center

The PBW theorem PBW gives an ordered monomial basis for the enveloping algebra identifies $\mathrm{gr}U(\mathfrak{g})$ with $S(\mathfrak{g})$. Its symmetrization map $\mathrm{sym}(x_1\cdots x_m)=\frac{1}{m!}{\sum }_{\sigma \in S_m}{x}_{\sigma (1)}\cdots x_{\sigma (m)}$ is a filtration-preserving vector-space isomorphism whose associated graded map is the identity. Because the adjoint action is a derivation on both sides, $\mathrm{sym}$ is $\mathfrak{g}$-equivariant. It therefore restricts to a filtered vector-space isomorphism $S(\mathfrak{g}{)}^{\mathfrak{g}}\stackrel{\sim }{\to }U(\mathfrak{g}{)}^{\mathfrak{g}}=Z(U(\mathfrak{g})),$ where the last equality holds because $\mathfrak{g}$ generates $U(\mathfrak{g})$. Consequently $\mathrm{gr}Z(U(\mathfrak{g}))=S(\mathfrak{g}{)}^{\mathfrak{g}}\mathrm{.}$

Fix a finite ordered basis of $\mathfrak{g}$ and order its monomials degree by degree. In each finite-dimensional space $H_d\subseteq S^d(\mathfrak{g})$, row reduction in this fixed monomial order specifies a basis; taking the union over $d\ge 0$ gives a homogeneous basis $(h_j)$ of $H$ without a simultaneous choice. Define its PBW lifts by $a_j=\mathrm{sym}(h_j)$. The PBW filtration is nonnegative and exhaustive, as is its restriction $F_{n}Z=Z\cap F_{n}U$ to the center. The symmetrization in step 1.1 places $a_j$ in filtration degree $d_j=\mathrm{deg}{h}_j$ and gives it leading symbol $h_j$.

By [A1, F1] and step 1.1, the symbols $(h_j)$ are a homogeneous free basis of $\mathrm{gr}U(\mathfrak{g})=S(\mathfrak{g})$ over $\mathrm{gr}Z(U(\mathfrak{g}))=S(\mathfrak{g}{)}^{\mathfrak{g}}$. The filtrations and specified lifts in step 2.1 meet the hypotheses of Filtered freeness lifts from associated graded algebras. Its finite-degree induction lifts the graded expansion of each element of $U(\mathfrak{g})$, while its highest-degree argument proves independence; it follows that $(a_j)$ is a left and right basis over the center.