Exercise 1, § 8

Let $f$ be an invariant polynomial function on $\mathfrak{g}$. Show that $f$ is invariant under Aut($\mathfrak{g}$) if and only if $f|\mathfrak{h}$ is invariant under Aut(R). Deduce that, if the Dynkin graph of R has a non-trivial automorphism, there exists an invariant polynomial function on $\mathfrak{g}$ that is not invariant under Aut($\mathfrak{g}$).