Exercise 17, § 8

Assume that $k$ is algebraically closed, and put G = Aut$_e(\mathfrak{g})$. Let $x, y\in \mathfrak{g}$. Show the equivalence of:

(i) The semi-simple components of $x$ and $y$ are G-conjugate.

(ii) For every invariant polynomial function $f$ on $\mathfrak{g}$, we have $f(x) =f(y)$.

(Use Exerc. 15 and 16.)