The statement I'm interested in (ambiguously named Jordan's lemma) concerns the setting where knowing that $A^2 = B^2 = I$ for two unitary matrices, for instance, ensures a decomposition of the vector space on which $A, B$ act into $A, B$-invariant subspaces with block size at most 2.
I'm interested in understanding how group-relations among unitary representations of finite groups affect the decomposition of an underlying vector space on which these representations act into invariant subspaces. This post mentions some of these topics, but does not provide strong references. What are good treatments (e.g., textbooks)?