Dimension of $\operatorname{End}_{\mathbb C} \mathbb H$ as $\mathbb {R}$ vector space.

Question

On page 15 of this note

$\operatorname{End}_{\mathbb C} \mathbb H$ is an $8$-dimensional real vector space.

Is there a simple way to see this?

Answer 1

$\mathbb H$ is a $2$-dimensional $\mathbb C$ vector space, so $\operatorname{End}_\mathbb C(\mathbb H)\cong M_2(\mathbb C)$. The latter is, of course, is $8$ dimensional considered as an $\mathbb R$ vector space.