Exercise 16, § 8

Let $\mathfrak{a}$ be a semi-simple Lie algebra, $x$ an element of $\mathfrak{a}$, and $x_s$ the semi-simple component of $x$. Show that, if $f$ is an invariant polynomial function on $\mathfrak{a}$, then $f(x) =f(x_s)$. (Reduce to the case in which $f$ is of the form $x \rightarrow$ Tr $\rho (x)^n.$)