Proposition 4

Assume that $\rho$ is irreducible. Then, $\rho$ is of real (resp. complex, resp. quaternionic) type if and only if the integral $\int_G$ Tr$\rho (g^2)dg$ is equal to 1 (resp. 0, resp. $-1$).

Denote by $\check{\rho}$ the contragredient representation of $\rho$ on $W^*$ (defined by $\check{\rho}(g) =^t\rho (g^{-1})$). Applying Lemma 2 to $\check{\rho}(g)$ and integrating over G gives

$\int_G$ Tr$\rho (g^2)dg=\int_G$ Tr $^t\rho (g^{-2})dg=\int_G$ Tr $\mathbf{S}^2( \check{\rho}(g))dg-\int_G$ Tr $\wedge \wedge^2( \check{\rho}(g))dg$

hence, by Lemma 1,

$\int_G$ Tr $\rho (g^2)dg=$ dim($\mathbf{S}^2W^*$)$^G-$ dim($\wedge \wedge^2W^*$)$^G$.

But $\mathbf{S}^2W^*$ (resp. $\wedge \wedge^2W^*$) can be identified with the space of symmetric (resp. alternating) bilinear forms on W. Thus, the proposition follows immediately from Prop. 3.