Exercise 14, § 8

Let $\mathfrak{a}$ be a semi-simple Lie algebra, and $Z(\mathfrak{a})$ the centre of $U(\mathfrak{a})$. Show that $U(\mathfrak{a})$ is a free $Z(\mathfrak{a}$)-module. (Remark that gr $U(\mathfrak{a})$ is isomorphic to $\mathbf{S}(\mathfrak{a})$ and $\mathbf{S}(\mathfrak{a}^*)$, and use Remark 2 of no. $3.$)\*