 Solution

The cyclic products obtained by direct matrix multiplication are ${\sigma }_x{\sigma }_y=i{\sigma }_z$, ${\sigma }_y{\sigma }_z=i{\sigma }_x$, and ${\sigma }_z{\sigma }_x=i{\sigma }_y$. Reversing any two distinct indices changes the sign; equal-index products are $I$. Therefore every pair is covered byHere $\{A,B\}$ is the anticommutator and $[A,B]$ the commutator. Explicitly the three diagonal anticommutators are $2I$, all six off-diagonal ones are zero, and all three diagonal commutators vanish. The cyclic commutators are $2i{\sigma }_z,2i{\sigma }_x,2i{\sigma }_y$, respectively, with their reversed partners negative. The convention is ${ϵ}_{xyz}=1$.