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.



Are finite-dimensional real representations of semisimple real Lie algebras completely reducible?
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*.


