History
Adolf Hurwitz had shown how integration over a compact Lie group could be used to construct invariants, in the cases of unitary groups and compact orthogonal groups. Issai Schur in 1924 showed that this technique applied to show complete reducibility of representations for such groups via the construction of an invariant inner product. Weyl extended Schur's method to complex semisimple Lie algebras by showing they had a compact real form.
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