Exercise 19, § 7

We recall the notations of Prop. 10. Let $\overline{k}$ be an algebraic closure of $k$. If $x\in \mathfrak{g}$, denote by $\mathscr{X}_E(x)$ (resp. $\mathscr{X}_F(x)$, resp. $\mathscr{X}_G(x)$) the set of eigenvalues of $x_E$ (resp. $x_F$, resp. $x_G$) in $\overline{k}$. Show that $\mathscr{X}_G(x) =\mathscr{X}_E(x) +\mathscr{X}_F(x)$ for all $x\in \mathfrak{g}$ and that, when E and F are given, this property characterizes the simple $\mathfrak{g}$-module G up to isomorphism.