TITLE: Reference for representations of quaternion group
QUESTION [8 upvotes]: 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 $\mathbb{R}$.

One-dimensional representations that factor through $(\mathbb{Z}/2)^2$.
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.

REPLY [2 votes]: Here is a stupid proof which, I think, would not elicit any further questions:
Let $\left[1\right]$, $\left[-1\right]$, $\left[i\right]$, $\left[-i\right]$, $\left[j\right]$, $\left[-j\right]$, $\left[k\right]$, $\left[-k\right]$ be the elements of $Q$ which correspond to $1$, $-1$, $i$, $-i$, $j$, $-j$, $k$, $-k$ under the canonical embedding $Q\subseteq \mathbb H^{\times}$. Then, the $\mathbb R$-linear map $\mathbb H\oplus \mathbb R\oplus \mathbb R\oplus \mathbb R \oplus \mathbb R$ which sends every $\left(a+bi+cj+dk,e,f,g,h\right)$ to
$\dfrac{\left[1\right]-\left[-1\right]}{2}\left(a+b\left[i\right]+c\left[j\right]+d\left[k\right]\right) $
$+ \dfrac{\left[1\right]+\left[-1\right]}{2}\left(\left(e+f+g+h\right)\left[1\right]+\left(e-f-g+h\right)\left[i\right]+\left(e+f-g-h\right)\left[j\right]+\left(e-f+g-h\right)\left[k\right]\right)$
(for all reals $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$) is easily seen to be an $\mathbb R$-algebra isomorphism. Now, the representation theory of a direct product of algebras is well-known, and so is the representation theory of fields.