# Claim in Atiyah and Bott and Shapiro's paper on Clifford modules.

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.

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Here we let $Q(x) = -x^2 - y^2$ and $Q'(x) = x^2 + y^2$, so that $$C_2 = \mathrm{C}\ell(\mathbb{R}^2, Q)$$ and $$C^\prime_2 = \mathrm{C}\ell(\mathbb{R}^2, Q').$$
Let $\{e_1, e_2\}$ be an orthornomal basis for $(\mathbb{R}^2,Q)$. Then $C_2$ has generators $\{1, e_1, e_2, e_1 e_2\}$ satisfying the relations $$e_1^2 = e_2^2 = -1,$$ $$e_1 e_2 = -e_2 e_1.$$ Then the map $C_2 \longrightarrow \mathbb{H}$ defined on generators by $1 \mapsto 1$, $e_1 \mapsto i$, $e_2 \mapsto j$, and $e_1 e_2 \mapsto k$ is seen to be the desired isomorphism.
Now let $\{e_1^\prime, e_2^\prime\}$ be a basis for $(\mathbb{R}^2, Q')$. Then $C_2^\prime$ has generators $\{1, e_1^\prime, e_2^\prime, e_1^\prime e_2^\prime\}$ satisfying the relations $$(e_1^\prime)^2 = (e_2^\prime)^2 = 1,$$ $$e_1^\prime e_2^\prime = -e_2^\prime e_1^\prime.$$ Our desired isomorphism is given by $$C_2^\prime \longrightarrow \mathbb{R}(2),$$ $$a + be_1^\prime + ce_2^\prime + de_1^\prime e_2^\prime \mapsto \begin{pmatrix} a + b & c + d \\ c - d & a - b \end{pmatrix}.$$