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 $\mathrm{gr}Z(U(\mathfrak{g}))=S(\mathfrak{g}{)}^{\mathfrak{g}}\cong S(\mathfrak{h}{)}^W,$ the last isomorphism being Chevalley restriction for symmetric invariants.

Choose PBW lifts of a homogeneous basis of the harmonic space $H$ from [F1] to a subspace $\tilde{H}\subseteq U(\mathfrak{g})$.

The multiplication map $Z(U(\mathfrak{g}))\otimes \tilde{H}\to U(\mathfrak{g})$ has associated graded equal to the isomorphism in [F1], by step 1.1 and the chosen leading symbols in step 2.1. Filtered-graded comparison therefore makes multiplication an isomorphism of left $Z(U(\mathfrak{g}))$-modules. Since the center is central, the same basis gives a right-module isomorphism. Hence $U(\mathfrak{g})$ is free on both sides over its center.