Simply put, I need to find all primitive idempotents of the Clifford Algebra $Cl(1,3)$ over the complex numbers. I have found some general results but they're only applicable over the real numbers. I come from a physics background so I'm pretty much doing this by hand and checking if I haven't missed any cases.
So, defining the basis vectors as $\gamma_0$, $\gamma_1$, $\gamma_2$ and $\gamma_3$, such that $(\gamma_0)^2 = 1 = -(\gamma_1)^2 = - (\gamma_2)^2 = -(\gamma_3)^2$, all the primitive idempotents I have found thus far are \begin{align} P_1 & = \left({1+\gamma_0 \over 2}\right) \left({1-i \gamma_1 \gamma_2 \over 2}\right) \\ P_2 & = \left({1+\gamma_0 \over 2}\right) \left({1-i \gamma_2 \gamma_3 \over 2}\right) \\ P_3 & = \left({1+\gamma_0 \over 2}\right) \left({1-i \gamma_3 \gamma_1 \over 2}\right) \\ P_4 & = \left({1+\gamma_0 \over 2}\right) \left({1-i \gamma_0 \gamma_1 \gamma_2 \gamma_3 \over 2}\right) \end{align} and all variations with different signs inside the parentheses. How can I be sure that I haven't missed any?
Thank you very much for any help!