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$\DeclareMathOperator\Cl{Cl}$Let $V$ be a real vector space of dimension $p+q$. Let $Q$ be a non-degenerate quadratic form on $V$ of signature $(p,q)$ where $p$ is the number of positive eigenvalues, and $q$ is the number of negative ones.

The Clifford algebra $\Cl(p,q)$ is the quotient of the tensor algebra $\oplus_{k=0}^\infty V^{\otimes k}$ modulo the two sided ideal generated by elements $v^2-Q(v)$ for any $v\in V$.

I am looking for a classification up to an isomorphism of $\Cl(p,q)$. The cases $p=1,q=3$ and $p=3,q=1$ are of special interest. A reference would be helpful.

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The classification is described in quite some detail in the Wiki on the topic.

In particular, the case $(p,q)=(1,3)$ is isomorphic to $M_2(\mathbb H)$ and the case $(p,q)=(3,1)$ is $M_4(\mathbb R)$.

Based on what you've said, it looks like this is what you're looking for.

It should not be hard to turn up useful matter with details online.

E.g. https://empg.maths.ed.ac.uk/Activities/Spin/Lecture2.pdf

https://ncatlab.org/nlab/show/Clifford+algebra

I'm afraid I don't have a specific text to recommend, but I would be interested in seeing one...

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