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For questions about spin manifolds, the groups $\operatorname{Spin}(n)$, as well as generalisations such as $\operatorname{Pin}^{\pm}(n)$ and $\operatorname{Spin}^c(n)$. This tag should also be used for any questions about the geometry of spin manifolds, including questions involving Dirac operators and the Lichnerowicz formula.
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Triality of Spin(8)
Here's my favorite way to answer your question. Hopefully the answer to Robert Bryant's question is "yes".
Let $A$ be the ring of octonions (the "nonsplit" octonions over ${\mathbb R}$); it comes wi …