Simultaneous diagonalization of the tensor products of Dirac gamma matrices

Let $$\gamma_i\ (i=1,2,\ldots N)$$ be the Dirac gamma matrices satisfying the Clifford algebra $$\gamma_i\gamma_j+\gamma_j\gamma_i=2\delta_{ij} I\ \ (i,j=1,2,\ldots,N).$$ Then the tensor products $$\gamma_i\otimes\gamma_i$$ commute with each other: $$[\gamma_i\otimes\gamma_i, \gamma_j\otimes\gamma_j]=0.$$ This means all of $$\gamma_i\otimes\gamma_i$$ can be simultaneously diagonalized by some similarity transformation $$S$$. I wonder what the characterizations of such diagonalization are, such as the arrangement of eigenvalues of $$\gamma_i\otimes\gamma_i$$ after diagonalization, the (fast) computational algorithm for $$S$$ etc. I find surprisingly little about this elementary construction online, any reference will also be appreciated.

Update: I have basically solved problem, I will write an answer to it in the future if time permits.

• This reminds me the fact that in the exterior algebra $\Lambda(k^n)$, the sub-algebra generated by the elements $x\wedge y$ is a commutative ring. This is an ingredient of the proof of the theorem of Amitsur and Levitski. Apr 5 '19 at 16:50