Proposition 9

Let $\mathfrak{h}$ be a splitting Cartan subalgebra of $\mathfrak{g}$.

(i) The group $f(T_Q)$ is closed in Aut($\mathfrak{g}$) in the Zariski topology.

(ii) The group $f(q(T_P))$ is dense in $f(T_Q)$ in the Zariski topology.

Assertion (i) follows from the equality $f(T_Q) =$ Aut($\mathfrak{g},\mathfrak{h}$)$\cap$Ker$\varepsilon$ (Prop. 2). Put $m= (P(R) : Q(R))$. Let F be a polynomial function on V; we assume that F vanishes on the $m$th power of every element of $f(T_Q)$, and show that $F|f(T_Q) = 0$; in view of Lemma 3, this will prove (ii).

The set $V'$ of elements of V inducing the identity on $\mathfrak{h}$ and leaving each $\mathfrak{g}^{\alpha}$ stable can be identified with $k^R$. Let $F'$ be the restriction of F to $V'=k^R$; this is a polynomial function. We have $f(T_Q)\subset V'$. Let $B = (\alpha_1, . . . , \alpha_l)$ be a basis of R. For all $t= (t_1, . . . , t_l)\in k^{*B}$, let $\varphi (t)$ be the homomorphism from Q(R) to the group $k^*$ that extends $t$. Then $F'(f(\varphi (t)))$ can be written as a finite sum

By assumption,

for all $t_1, . . . , t_l\in k^*$. The $c_{n_1,...,n_l}$ are thus the coefficients of a polynomial in $l$ variables which vanishes on $k^{*l}$; hence they are all zero.