To satisfy a referee, I need a reference for the following well-known fact (which is not hard to prove, but it seems silly to prove it in a paper). I can't find it in either Serre or Fulton-Harris's books on representation theory. Can anyone provide me a reference?


Let $Q$ be the $8$-element quaternion group, so we have a central extension
$$1 \longrightarrow \mathbb{Z}/2 \longrightarrow Q \longrightarrow (\mathbb{Z}/2)^2 \longrightarrow 1.$$
Then the following constitute a complete list of irreducible representations of $Q$ over $\mathbb{R}$.


1. One-dimensional representations that factor through $(\mathbb{Z}/2)^2$.
2. A four-dimensional representation $W$ obtained from the left action of $Q$ on the real quaternions (viewed as a $4$-dimensional real vector space).


I remark that the representation $W$ is irreducible but not absolutely irreducible -- when we extend the field of scalars to $\mathbb{C}$, it breaks up into two copies of the (unique) two-dimensional irreducible complex representation.



Reference for representations of quaternion group
Since $Q\_8$ is an extraspecial group, if you want you can say that the result follows from Quillen's classification of their real representations in


Quillen, The mod 2 cohomology rings of extra-special 2-groups and the spinor groups


Mathematische Annalen, 1971


