Unitarization of a compact-group representation

For a compact group with normalized Haar measure, averaging any positive-definite Hermitian form,produces an invariant positive-definite Hermitian form.

For a continuous real representation $\rho :G\to GL(V)$ of a compact group, average any positive-definite real inner product against normalized Haar measure:The result is positive definite and $G$-invariant. An orthonormal basis for it conjugates the representation into an orthogonal group.