Properties of Clifford Algebras

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.

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en.wikipedia.org/wiki/Clifford_algebra - scroll down to the part about "structure". –  S. Carnahan Oct 4 '12 at 11:02
If you are interested in Clifford algebras, I suggest reading "Clifford Modules", of Atiyah, Bott and Shapiro. It's a masterpiece. –  Angelo Oct 4 '12 at 12:38