= Solution
The <Pauli-Lubanski pseudovector> and the spinor Lorentz transformation give
$$
[W_\mu,Q_\alpha]=i(\sigma_{\mu\nu})_\alpha{}^\beta P^\nu Q_\beta.
$$
Combining this with part iii and the <Pauli matrix> identity $\sigma^\nu\bar\sigma_\mu=\eta^\nu{}_\mu\mathbf1+2i\sigma^\nu{}_\mu$ cancels the term containing $\sigma_{\mu\nu}P^\nu$. In these conventions,
$$
\boxed{[B_\mu,Q_\alpha]=-\frac12P_\mu Q_\alpha}.
$$
Simultaneously reversing the convention for $W_\mu$ reverses the final sign, but the proportionality to $P_\mu Q_\alpha$ is invariant and is what the next part needs.
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