(I am using a throwaway account because I plan on possibly pointing to this answer in a referee report I am writing, and using my main account would be a bit too obvious.)
Let $\mathfrak{g}$ be a split semisimple real Lie algebra. Then it is a fact that all finite-dimensional irreducible representations of $\mathfrak{g}$ are of real type. This can be checked, for example, on a case-by-case basis by looking at the table on page 292 in Onishchik and Vinberg, Lie groups and algebraic groups (which however has numerous typos, see last page of Ran Cui, The Real-Quaternionic Indicator for corrections).
But of course this is not very satisfactory. Is there a simple direct argument that I have overlooked?
