It is a well known fact that Clifford algebras, $Cl(p,q)$, have similar properties depending on $(p-q)\mod 8$.

In most of the places I have found a proof of the theorem, explicit representations of the generators of the algebra are used. As an example, in physics literature, the Dirac matrices (representing the generators of $Cl(3,1)$) are constructed from the Pauli matrices (generators of $Cl(2,0)$).

**Question**

How could be the theorem demonstrated without the use of explicit representations? (if possible)

Thank you.