I've been considering the Clifford algebra over $\mathbb{R}$. For notation, I denote 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.
I'm aware of the basic isomorphisms $C_1\approx\mathbb{C}$ and $C'_1\approx\mathbb{R}\times\mathbb{R}$ and the like. I was looking further into one of my reference books though, and there are the more general isomorphisms $ C_{n+2}\approx C'_n\otimes_\mathbb{R} C_2;\qquad C'_{n+2}\approx C_n\otimes_\mathbb{R} C'_2. $
No proof is offered though, and I don't see how these isomorphisms are established. Is there a more in depth reference where these are exhibited, or even a proof or sketch here? Cheers.