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.

compactreal semisimple Lie algebra and false in other cases. $\endgroup$2more comments