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in Schwartz' QFT book it says (p. 483 - 484)

In Problem 25.3 this is repeated asking the reader for a proof. I wonder though if this is really true. I know this can be proven for Lie algebras of compact Lie groups (or to be precise, every representation is equivalent to a hermitian one), but I would not know how to show it for semisimple Lie algebras.A theorem that explains the importance of semisimple Lie algebras in physics states that all finite-dimensional representations of semisimple algebras are Hermitian (see Problem 25.3