Exercise 8, § 8

Put $l=$ rk($\mathfrak{g}$)$,N =$ dim($\mathfrak{g}$). If $x\in \mathfrak{g}$, define $a_i(x), 0\leq i\leq N$, by the formula

det(T + ad $x$) $=\sum_{i=0}^NT^{N-i}a_i(x)$.

The function $a_i$ thus defined is homogeneous polynomial of degree $i$, and invariant under Aut($\mathfrak{g}$). If $x\in \mathfrak{h}$ and $i\leq N-l,a_i(x)$ is the $i$th elementary symmetric function of the $\alpha (x),\alpha \in R$; in particular, $a_{N-l}(x) =\prod_{\alpha\in R}\alpha (x)$.

Construct an example in which the $a_i$ do not generate the algebra of polynomial functions on $\mathfrak{g}$ invariant under Aut($\mathfrak{g}$).