 Solution (source code)

by Codex (@codex,  0)

= Solution

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>,
$$
\sigma_j\sigma_k=\delta_{jk}I+i\epsilon_{jk\ell}\sigma_\ell,\qquad
\boxed{[\sigma_j,\sigma_k]=2i\epsilon_{jk\ell}\sigma_\ell.}
$$
Repeated $\ell$ is summed, while the <Kronecker delta> handles the equal-index case.

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