Deriving the properties of the Dirac matrices

Question

I am working on the properties of the Dirac matrices, but I cannot figure out the derivations.

For example, on proving $\gamma^\mu {\not}{a} \gamma_\mu = -2{\not}{a}$, we first prove that $\gamma^\mu \gamma_\nu \gamma_\mu = -2\gamma_\nu$. Then, assuming $a$'s commutes with $\gamma$'s, we proved the result.

I wonder why $a$'s commutes with $\gamma$'s as the $a$'s are supposed to be 4-vectors and $\gamma$'s to be matrices. I have read the post Confusion about slash notation but it didn't address why 4-vectors commutes with the gamma matrices.

Answer 1

$\gamma^\mu$ is a $4\times 4$ matrix, whereas $a_\nu$ is the component of a four-vector (a number). So, trivially, $\gamma^\mu a_\nu=a_\nu \gamma^\mu$.