Let $G\subset GL(n,\mathbb C)$ be an algebraic group with Lie algebra $\mathfrak g$. Let $A\in \mathfrak g$ and $A=S+N$ be its additive Jordan decomposition (A is a diagonalizable matrix and N is nilpotent).
For a smooth function $f$ on $G$, we consider the vector field $X_A$ defined by
$$X_A f(g):=\frac{d}{dx}\bigg\vert_{t=0} f(u\exp(tA)).$$
Now we have by linearity of $A\mapsto X_A$ that $X_A=X_S+X_N$.
Is it true that there exists a polynomial $\psi$ such that $X_S=\psi(X_A)$?
This question rises from the book Symmetry, Represenation and Invariants by Goodman and Wallach, page 59 at the end of the proof for Theorem 1.6.5. The question asked here is a bit more general than the argument over there. It will also be appreciated if anyone could explain his argument there to me, thanks!