Suppose $\mathfrak{g}$ is a real form of a semisimple Lie algebra $\mathfrak{g}_\mathbb{C} = \mathfrak{g} \otimes_\mathbb{R} \mathbb{C}$. Then we have the following:
- There is an equivalence of monoidal categories between the category of finite-dimensional complex representations of $\mathfrak{g}$ and the category of finite-dimensional complex representations of $\mathfrak{g}_\mathbb{C}$.
- One can classify the irreducible real representations of $\mathfrak{g}$. These are either restrictions to $\mathfrak{g}$ of irreducible complex representations of $\mathfrak{g}_\mathbb{C}$ (which remain irreducible over $\mathfrak{g}$) or real forms of irreducible complex representations of $\mathfrak{g}_\mathbb{C}$ (in this case there is a real structure on the underlying space of the representation that commutes with the action of $\mathfrak{g}$). See, for example, Theorem 1 on page 65 of Onishchik, Lectures on Real Semisimple Lie Algebras.
What I'd like to know is if the category of finite-dimensional real representations of $\mathfrak{g}$ is semisimple. In other words, is every finite-dimensional real representation of $\mathfrak{g}$ completely reducible? Despite spending quite some time searching through the literature, I can't seem to find the answer to this question.
Best Answer
As pointed out by YCor in the comments, the answer is Yes. A reference has been pointed out to me: Chapter III, Section 7, Theorem 8 in Jacobson's book Lie algebras.