Let $(\pi, V)$ be an irreducible unitary representation of a compact group $G$ with Haar measure $dg$. I would like to prove the following identity for $v_i \in V$:
$$\int_G\langle \pi(g)v_1|v_2\rangle \overline{\langle \pi(g) v_3 | v_4 \rangle} = \frac{1}{\dim V}\langle v_1 | v_2 \rangle \overline{\langle v_3 | v_4 \rangle}$$
I just started learning about topological representations and I feel a bit out of my depth here. The only identity I can think of that might be useful is $\langle v | v' \rangle = \int_G \langle \pi(g)v|\pi(g)v' \rangle dg$, but I don't know how I would apply it.
Hints are much appreciated.