Weyl's unitary trick

Summary

Weyl's unitary trick proves facts about finite-dimensional representations of complex semisimple Lie groups and Lie algebras. It does so by passing to a compact real form. Representations of the compact form can be averaged over the group, and this gives results such as complete reducibility. The trick then carries the results back, because the complex Lie algebra and its compact real form have the same finite-dimensional complex representations.