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.