Exercise 6, § 7

Let $\varphi$ be an involutive automorphism of $\mathfrak{g}$ whose restriction to $\mathfrak{h}$ is $-$Id. Show that, if E is a $\mathfrak{g}$-module, there exists a non-degenerate symmetric bilinear form $\Psi$ on E such that

$\Psi (x.a, b) +\Psi (a, \varphi (x).b) = 0$ for $x\in \mathfrak{g}, a, b\in E$.

(Reduce to the case in which E is simple. Show that the transform of E by $\varphi$ is isomorphic to the dual $E^*$ of E, and deduce the existence of a non-degenerate bilinear form $\Psi$ satisfying the conditions above. Show next that, if $e$ is a primitive vector in E, then $\Psi (e, e)\not= 0$. Deduce that $\Psi$ is symmetric.)