Struggling with connection between Clifford Algebra (/GA) and their matrix generators As I thought I understood things, the Gamma matricies behave as the 4 orthogonal unit vectors of the Clifford algebra $\mathcal{Cl}_{1,3}(\mathbb C)$, (also the Pauli matricies are for the 3 of $\mathcal{Cl}_{3}(\mathbb C)$??).
But, I'm not getting results that are intuitive when I perform algebra using multivectors built out of these.
The dot and wedge products are $A\cdot B=\frac{1}{2}(AB+BA)$ and $A\wedge B=\frac{1}{2}(AB-BA)$, respectively. But, I should be able to wedge multiply all 4 unit vectors in order to achieve the unit pseudovector, but the result is zero when I attempt to use these definitions to matrix multiply $\gamma_0\wedge\gamma_1\wedge\gamma_2\wedge\gamma_3$. 
I think I'm missing something large...

Those formulas for dot and wedge product are only valid when $A$ and $B$ are vectors, not if they are bivectors, trivectors, etc.
