Remark 1

\*Let $P_1, . . . ,P_l$ be algebraically independent homogeneous elements of I that generate the algebra I. Then $(P_1, . . . ,P_l)$ is an $\mathbf{S}(\mathfrak{a}^*)$-regular sequence (Chap. V, §5, no. 5). Indeed, extending the base field if necessary, we can assume that $\mathfrak{a}=\mathfrak{g}$ is splittable. Now let N = dim $\mathfrak{g}$, and let

be a basis of the orthogonal complement of $\mathfrak{h}$ in $\mathfrak{g}^*$. Let $\mathfrak{m}$ be the ideal of $\mathbf{S}(\mathfrak{g}^*)$ generated by $P_1, . . . ,P_l,Q_1, . . . ,Q_{N-l}$. Then $\mathbf{S}(\mathfrak{g}^*)\mathfrak{m}$ is isomorphic to $\mathbf{S}(\mathfrak{h}^*)/J$, where J is the ideal of $\mathbf{S}(\mathfrak{h}^*)$ generated by $i(P_1), . . . , i(P_l)$. By Th. 1 and Chap. V, §5, no. 2, Th. $2,\mathbf{S}(\mathfrak{h}^*)/J$ is a finite dimensional vector space, and hence so is $\mathbf{S}(\mathfrak{g}^*)/\mathfrak{m}$. By a result of Commutative Algebra, it follows that $(P_1, . . . ,P_l,Q_1, . . . ,Q_{N-l})$ is an $\mathbf{S}(\mathfrak{g}^*$)-regular sequence, and a fortiori so is $(P_1, . . . ,P_l)$.

$^3$ It can be shown (B. KOSTANT, Lie group representations on polynomial rings,

Amer. J. Math., Vol. LXXXV (1963), pp. 327-404, Th. 10 and 15) that $\mathfrak{p}$ is a

prime ideal of $\mathbf{S}(\mathfrak{a}^*)$ and that $\mathbf{S}(\mathfrak{a}^*)/\mathfrak{p}$ is integrally closed.