Corollary 2

Let B be a basis of R, $R_+$ (resp. $R_-$) the set of positive (resp. negative) roots of $(\mathfrak{g},\mathfrak{h})$ relative to B$,\mathfrak{n}_+=\sum_{\alpha\in R_+}\mathfrak{g}^{\alpha},\mathfrak{n}_-=\sum_{\alpha\in R_-}\mathfrak{g}^{\alpha}$, $\mathbf{S}(\mathfrak{h})$ the symmetric algebra of $\mathfrak{h}$, and J the ideal of $\mathbf{S}(\mathfrak{g})$ generated by $\mathfrak{n}_+\cup \mathfrak{n}_-$.

(i) $\mathbf{S}(\mathfrak{g}) =\mathbf{S}(\mathfrak{h})\oplus J$.

(ii) Let $j$ be the homomorphism from the algebra $\mathbf{S}(\mathfrak{g})$ to the algebra $\mathbf{S}(\mathfrak{h})$ defined by the preceding decomposition of $\mathbf{S}(\mathfrak{g})$. Let $I(\mathfrak{g})$ be the set of invariant elements of $\mathbf{S}(\mathfrak{g})$. Let $\mathbf{S}(\mathfrak{h})^W$ be the set of elements of $\mathbf{S}(\mathfrak{h})$ invariant under the operation of W. Then $j|I(\mathfrak{g})$ is an isomorphism from $I(\mathfrak{g})$ to $\mathbf{S}(\mathfrak{h})^W$.

Assertion (i) is clear. The Killing form defines an isomorphism from the vector space $\mathfrak{g}^*$ to the vector space $\mathfrak{g}$, which extends to an isomorphism $\xi$ from the $\mathfrak{g}$-module $\mathbf{S}(\mathfrak{g}^*)$ to the $\mathfrak{g}$-module $\mathbf{S}(\mathfrak{g})$. We have $\xi (I(\mathfrak{g}^*)) = I(\mathfrak{g})$. The orthogonal complement of $\mathfrak{h}$ with respect to the Killing form is $\mathfrak{n}_++\mathfrak{n}_-($§2, no. 2, Prop. 1). If we identify $\mathfrak{h}^*$ with the orthogonal complement of $\mathfrak{n}_++\mathfrak{n}_-$ in $\mathfrak{g}^*$, then $\xi (\mathfrak{h}^*) =\mathfrak{h}$, so $\xi (\mathbf{S}(\mathfrak{h}^*)) =\mathbf{S}(\mathfrak{h})$ and $\xi (\mathbf{S}(\mathfrak{h}^*)^W) =\mathbf{S}(\mathfrak{h})^W$. Finally, $\xi^{-1}(J)$ is the set of polynomial functions on $\mathfrak{g}$ that vanish on $\mathfrak{h}$. This proves that $\xi$ transforms the homomorphism $i$ of Th. 1 into the homomorphism $j$ of Cor. 2. Thus assertion (ii) follows from Th. 1 (i).