Generalization of Schur's Lemma: finite dim. representations of real Lie algebras

Question

Let $V$ be an irreducible finite dimensional real representation of a real finite dimensional Lie algebra $\mathfrak{g}$. From Schur's Lemma, what is $Hom_\mathfrak{g}(V,V)$ or $End_\mathfrak{g}(V,V)$?

I am getting confused with the terminology employed in the literature. Do we have a (real?) (non?) associative (division?) ring/algebra/algebraic extension?

Many thanks in advance for this clarification.

Answer 1

Note that $End_{\mathfrak{g}}(V)$ is an associative subalgebra of $End(V)$; and it is a real division algebra by Schur's lemma. The real division algebras of finite dimension are $\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$, but $\mathbb{O}$ is not associative.