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.