In an earlier answer, rschwieb kindly pointed me in the direction of Bott periodicity. Just out of curiosity I was reading through a paper on periodicity of Clifford algebras. There was a list of isomorphisms, "all of them easy to prove according to the author," but the last one I couldn't really work out at all. I think it's pretty well known, the isomoprhism in question is $$ C_{n+8}\approx C_n\otimes_\mathbb{R}M_{16}(\mathbb{R}) $$ regardless of whether $C$ is the clifford algebra associated with a positive or negative definite form.
This isomorphism is in a lot of documents that popped up on google, but nowhere a satisfying proof. Does someone have one here?
As for notation, I'm denoting the Clifford algebras $C_n$ associated with the vector space $\mathbb{R}^n$ with negative definite form, and $C'_n$ associated with $\mathbb{R}^n$ with positive definite form, although from what I understand the isomoprhism is true in either case.
The argument I know, and which satisfies me is:
Show first that $C_{k+2}\cong C_k'\otimes C_2$ and $C_{k+2}'\cong C_k\otimes C_2'$ by exhibiting isomorphisms.
Notice that this implies that $C_4\cong C_2\otimes C_2'$.
Using the first point twice, $C_{k+4}\cong C_k\otimes C_2\otimes C_2'$ and, by the second point, this is $C_k\otimes C_4$.
Using this last point twice now, we see that $C_{k+8}\cong C_k\otimes C_4\otimes C_4$.
Finally, show by hand that $C_4\cong M_2(\mathbb H)\cong M_2(\mathbb R)\otimes\mathbb H$, and, using this, that $C_4\otimes C_4\cong M_2(\mathbb R)\otimes M_2(\mathbb R)\otimes\mathbb H\otimes\mathbb H\cong M_{16}(\mathbb R)$, because $\mathbb H\otimes\mathbb H\cong M_4(\mathbb R)$.