I'm working on trying to solve the following problem:
Using the following expressión for the square of Pauli-Lubanski vector:$$W^2=-\frac{1}{2}M_{\mu\nu}M^{\mu\nu}P_{\alpha}P^{\alpha}+M^{\mu\nu}M_{\lambda\nu}P_{\mu}P^{\lambda}$$ show that $$[W^2,P_{\sigma}]=0$$ I know that there are other ways to prove the same result; for example, first showing that $[W_{\mu},P_{\nu}]=0$ but I have to use the formula that it gives to me, because It was deduced on the previous exercise. My calculations are like following; first of all $$[W^2,P_{\sigma}]=-\frac{1}{2}[M_{\mu\nu}M^{\mu\nu}P_{\alpha}P^{\alpha},P_{\sigma}]+[M^{\mu\nu}M_{\lambda\nu}P_{\mu}P^{\lambda},P_{\sigma}]$$ but, using the properties of conmutator, the conmutation relations of generators of Poincaré Algebra and renaming indexes I obtain, for the first conmutator on the right side, the following: $$[M_{\mu\nu}M^{\mu\nu}P_{\alpha}P^{\alpha},P_{\sigma}]=(M_{\mu\nu}[M^{\mu\nu},P_{\sigma}]+[M_{\mu\nu},P_{\sigma}]M^{\mu\nu})P_{\alpha}P^{\alpha}=i(M_{\mu\nu}\eta_{\sigma\lambda}(\eta^{\nu\lambda}P^{\mu}-\eta^{\nu\lambda}P^{\nu})+(\eta_{\nu\sigma}P_{\mu}-\eta_{\mu\sigma}P_{\nu})M^{\mu\nu})P_{\alpha}P^{\alpha}=2i(M_{\mu\sigma}P^{\mu}+\eta_{\nu\sigma}P_{\mu}M^{\mu\nu})P_{\alpha}P^{\alpha}$$ On the other hand, for the second conmutator on the right side I obtain the following: $$[M^{\mu\nu}M_{\lambda\nu}P_{\mu}P^{\lambda},P_{\sigma}]=[M^{\mu\nu}M_{\lambda\nu},P_{\sigma}]P_{\mu}P^{\lambda}=(M^{\mu\nu}[M_{\lambda\nu},P_{\sigma}]+\eta_{\sigma\epsilon}[M^{\mu\nu},P^{\epsilon}])P_{\mu}P^{\lambda}=(M^{\mu\nu}i(\eta_{\nu\sigma}P_{\lambda}-\eta_{\lambda\sigma}P_{\nu})+i\eta_{\sigma\epsilon}(\eta^{\nu\epsilon}P^{\mu}-\eta^{\mu\epsilon}P^{\nu})M_{\lambda\nu})P_{\mu}P^{\lambda}=i(\eta_{\nu\sigma}M^{\mu\nu}P_{\lambda}P_{\mu}P^{\lambda}+P^{\mu}M_{\lambda\sigma}P_{\mu}P^{\lambda})$$ The problem, accoding to me, comes when I try to compare both expressions using again the conmutation relations of Poincare algebra, because I obtain the following expression: $$\eta_{\nu\sigma}M^{\mu\nu}P_{\lambda}P_{\mu}P^{\lambda}+P^{\mu}M_{\lambda\sigma}P_{\mu}P^{\lambda}=\eta_{\nu\sigma}([M^{\mu\nu},P_{\mu}]+P_{\mu}M^{\mu\nu})P_{\lambda}P^{\lambda}+([P^{\mu},M_{\lambda\sigma}]+M_{\lambda\sigma}P^{\mu})P_{\mu}P^{\lambda}=\eta_{\nu\sigma}(i\eta_{\mu\alpha}(\eta^{\nu\alpha}P^{\mu}-\eta^{\mu\alpha}P^{\nu})+P_{\mu}M^{\mu\nu})P_{\lambda}P^{\lambda}+M_{\lambda\sigma}P_{\mu}P^{\mu}P^{\lambda}-i\eta^{\mu\alpha}(\eta_{\sigma\alpha}P_{\lambda}-\eta_{\lambda\alpha}P_{\sigma})P_{\mu}P^{\lambda}=(M_{\mu\sigma}P^{\mu}+\eta_{\nu\sigma}P_{\mu}M^{\mu\nu}-3iP_{\sigma})P_{\lambda}P^{\lambda}$$ I have tried for three complete days to eliminate the $3iP_{\sigma}$ term, but I couldn't. I have checked all my calculations, and made it many times using other ways, but I couldn't find where the mistake is, or what I'm doing wrong. Could anybody help me, please?. Am I not considering something? Note: In some steps I used that $M_{\mu\nu}=-M_{\nu\mu}$ :D .