In Weinbergs QFT book vol 1, pg. 60, equations (2.4.8) and (2.4.10) are respectively $$U(\Lambda,a)J^{\rho\sigma}U^{-1}(\Lambda,a)={\Lambda_\mu}^\rho {\Lambda_\nu}^\sigma(J^{\mu\nu}-a^\mu P^\nu+a^\nu P^\mu)$$and $$J^{\rho\sigma}+i[\frac{1}{2}w_{\mu\nu}J^{\mu\nu}-\epsilon_\mu P^\mu,J^{\rho\sigma}]=J^{\rho\sigma}{+w_\mu}^\rho J^{\mu\sigma}+ {w_\nu}^\sigma J^{\rho\nu} - \epsilon^\rho P^\sigma + \epsilon^\sigma P^\rho.$$ I added a term of $J^{\rho\sigma}$on both sides of eqn (2.4.10).
The second equation is derived from the first at by taking infinitesimal Lorentz transformations ${\Lambda^\mu }_\nu={\delta^\mu}_\nu + {w^\mu}_\nu$ and $a^\mu=\epsilon^\mu$.
I tried to prove this myself by taking the infinitesimal inverse Lorentz tranformation as $${({\Lambda^{-1}})^\mu}_\nu={\Lambda_\nu}^\mu={\delta^\mu}_\nu-{w^\mu}_\nu.$$
Substituting the above expression and $a^\mu=\epsilon^\mu$ into the RHS of eqn (2.4.8), I get $${\Lambda_\mu}^\rho {\Lambda_\nu}^\sigma(J^{\mu\nu}-a^\mu P^\nu+a^\nu P^\mu)=J^{\rho\sigma}-{w^\rho}_\mu J^{\mu\sigma}-{w^\sigma}_\nu J^{\rho\nu}- \epsilon^\rho P^\sigma + \epsilon^\sigma P^\rho.$$
Hence I will get the correct expression for the RHS of eqn (2.4.10) if $${w^\rho}_\mu=-{w_\mu}^\rho$$ $${w^\sigma}_\nu=-{w_\nu}^\sigma$$ How can I show that the above two expressions are true?
