Proposition 4

Let $\mathfrak{a}$ be a semi-simple Lie algebra, $l$ its rank. Let I (resp. $I'$) be the set of elements of $\mathbf{S}(\mathfrak{a}^*)$ (resp. $\mathbf{S}(\mathfrak{a})$) invariant under the representation induced by the adjoint representation of $\mathfrak{a}$. Let Z be the centre of the enveloping algebra of $\mathfrak{a}$.

(i) I and $I'$ are graded polynomial algebras (Chap. V, §5, no. 1) of transcendance degree $l$.

(ii) Z is isomorphic to the algebra of polynomials in $l$ indeterminates over $k$.

The canonical filtration of the enveloping algebra of $\mathfrak{a}$ induces a filtration of Z. By Chap. I, §2, no. 7, Th. 1 and p. 25, gr Z is isomorphic to $I'$. In view of Commutative Algebra, Chap. III, §2, no. 9, Prop. 10, it follows that (i) $=\Rightarrow$ (ii).

On the other hand, Th. 1 and its Cor. 2 show that (i) is true whenever $\mathfrak{a}$ is split. The general case reduces to that case in view of the following lemma: