Proposition 9

Let $\mathfrak{g}$ be a Lie algebra, $\mathfrak{g}'$ a subalgebra of $\mathfrak{g}$. Every element of $\mathfrak{g}'$ regular in $\mathfrak{g}$ is regular in $\mathfrak{g}'$.

For $x\in \mathfrak{g}'$, the restriction of ad$_{\mathfrak{g}}x$ to $\mathfrak{g}'$ is ad$_{\mathfrak{g}'}x$, and so defines an endomorphism $u(x)$ of the vector space $\mathfrak{g}/\mathfrak{g}'$ by passage to the quotient. Let $d_0(x)$ (resp. $d_1(x)$) be the dimension of the nilspace of ad$_{\mathfrak{g}'}(x)$ (resp. of $u(x)$), and let $c_0$ (resp. $c_1$) be the minimum of $d_0(x)$ (resp. $d_1(x)$) when $x$ belongs to $\mathfrak{g}'$. There exist non-zero polynomial maps $p_0, p_1$ from $\mathfrak{g}'$ to $k$ such that

Since $k$ is infinite, the set S of $x\in \mathfrak{g}'$ such that $d_0(x) =c_0$ and $d_1(x) =c_1$ is non-empty. Every element of S is regular in $\mathfrak{g}'$. On the other hand, S is the set of elements of $\mathfrak{g}'$ such that the nilspace of ad$_{\mathfrak{g}}x$ has minimum dimension, and thus contains every element of $\mathfrak{g}'$ regular in $\mathfrak{g}$.