Past exam of the mathematical methods I course of the University of Oxford / 2015 / 12 / i

Direct multiplication of the Pauli matrices gives ${\sigma }_j^2=I$, ${\sigma }_1{\sigma }_2=i{\sigma }_3$ and its cyclic permutations; reversing unequal factors changes the sign. Hence, using the Levi-Civita symbol,Repeated $\ell$ is summed, while the Kronecker delta handles the equal-index case.