Lemma 2

Let $\rho$ be a finite dimensional linear representation of $\mathfrak{a}$, and $m$ an integer $\geq 0$. The function $x \rightarrow$ Tr($\rho (x)^m$) on $\mathfrak{a}$ is an invariant polynomial function.

Put $g(x_1, . . . , x_m) =$ Tr($\rho (x_1). . . \rho (x_m)$) for $x_1, . . . , x_m\in \mathfrak{a}$. If $x\in \mathfrak{a}$, we have

$=\sum_{1\leq i\leq m}$ Tr($\rho (x_1). . . \rho (x_{i-1})[\rho (x), \rho (x_i)]\rho (x_{i+1}). . . \rho (x_m)$)

= Tr($\rho (x)\rho (x_1). . . \rho (x_m)$)$-$ Tr($\rho (x_1). . . \rho (x_m)\rho (x)$) $= 0$,

so $\theta^*(x)g= 0$. Let $h$ be the symmetric multilinear form defined by

$h(x_1, . . . , x_m) =m1$! $\sum g(x_{\sigma(1)}, . . . , x_{\sigma(m)})$.

$\sigma \in \mathfrak{S}_m$

For all $x\in \mathfrak{a}$, we have $\theta^*(x)h= 0$ and Tr($\rho (x)^m$) $=h(x, . . . , x)$, hence the lemma.