Exercise 6, § 8

Let $\mathfrak{a}$ be a Lie algebra, $f$ a polynomial function on $\mathfrak{a}$, and $x, y$ two elements of $\mathfrak{a}$. Put $f_y=\theta^*(y)f$ (cf. no. 3), and denote by $D_xf$ the tangent linear map of $f$ at $x$ (Chap. VII, App. I, no. 2). Show that $f_y(x) = (D_xf)([x, y])$. Deduce that $D_xf$ vanishes on Im(ad $x$) when $f$ is invariant.