In Atiyah, Shapiro, and Bott's paper on Clifford modules, they prove Proposition 4.2 on page 11 that there are isomorphisms $C_k\otimes_\mathbb{R} C_2^\prime\cong C_{k+2}^\prime$ and $C_k^\prime\otimes\mathbb{R} C_2\cong C_{k+2}$.
Immediately following the proof, they say it is clear that $C_2\cong\mathbb{H}$ and $C_2^\prime\cong\mathbb{R}(2)$. I get that $C_1\cong\mathbb{C}$ and $C_1^\prime\cong\mathbb{R}\oplus\mathbb{R}$, but those don't seem to be of use with the isomorphisms they proved.
Is there a quick explanation of how those isomorphisms for $C_2$ and $C_2^\prime$ are so easily seen? Thanks.