Invariant bilinear form as an intertwiner

A bilinear form $B$ on $V$ is invariant exactly when $v\mapsto B(v,-)$ is a $G$-homomorphism $V\to V^{\ast }$. On an irreducible representation, Schur's lemma makes a nonzero such form nondegenerate and unique up to scale.

Transposing a nonzero invariant bilinear form on an irreducible complex representation gives a scalar multiple $B^T=\lambda B$. Transposing twice yields ${\lambda }^2=1$, so the form is symmetric or alternating.