 Solution

The Pauli-Lubanski pseudovector and the spinor Lorentz transformation giveCombining this with part iii and the Pauli matrix identity ${\sigma }^{\nu }{\bar{\sigma }}_{\mu }={\eta }^{\nu }{}_{\mu }1+2i{\sigma }^{\nu }{}_{\mu }$ cancels the term containing ${\sigma }_{\mu \nu }{P}^{\nu }$. In these conventions,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.