It is well-known that any finite-dimensional complex representations of a compact real semisimple Lie algebra are unitarizable.
We can prove this from the fact that every finite-dimensional representation of a compact group is unitarizable by averaging with a Haar measure.
My questions is: Is there a "purely-algebraic" proof of this statement
without using any properties of Lie groups?
Does anyone know a reference or a idea?
I'm keeping in mind that Weyl used these facts to show complete reducibility of representations of complex semisimple Lie algebras and now algebraic proofs of complete reducility are known.