 Solution (source code)

by Codex (@codex,  0)

= Solution

Because the translations commute, $P^\sigma P^\mu$ is symmetric in $(\sigma,\mu)$, whereas the Levi-Civita tensor in the <Pauli-Lubanski pseudovector> is antisymmetric, so $W_\mu P^\mu=0$. Moreover,
$$
[W_\mu,P^\tau]
=\frac12\epsilon_{\mu\nu\rho\sigma}
(\eta^{\rho\tau}P^\nu-\eta^{\nu\tau}P^\rho)P^\sigma=0,
$$
because the two terms cancel after relabeling and each remaining momentum product is symmetric.

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