Proposition 3

Let $f$ be a polynomial function on $\mathfrak{g}$. The following conditions are equivalent:

(i) $f\circ s=f$ for all $s\in$ Aut$_e(\mathfrak{g})$;

(ii) $f\circ s=f$ for all $s\in$ Aut$_0(\mathfrak{g})$;

(iii) $f$ is invariant.

The equivalence of (i) and (iii) follows from formula (2) and Lemma 3. It follows from this that (iii) implies (ii) by extension of the base field. The implication (ii) $=\Rightarrow$ (i) is clear.

Note carefully that, if $f$ satisfies the conditions of Prop. $3,f$ is not in general invariant under Aut($\mathfrak{g}$) (Exerc. 1 and 2).