S. Weinberg, "The Quantum theory of fields: Foundations" (1995), Eq. 2.4.8 Unfortunately I'm struggling to understand how do we get eq. (2.4.8) from eq. (2.4.7), p. 60; namely how $(\Lambda \omega \Lambda^{-1} a)_\mu P^\mu$ is transformed into $\Lambda_\mu^{\;\rho}\Lambda_\nu^{\;\sigma}(-a^\mu P^\nu + a^\nu P^\mu)$. Why there are those two terms in the end, and not just $2a^\nu P^\mu$? I believe it must be connected with the antisymmetric nature of $\omega_{\mu\nu}$, but I'd be glad to see a rigorous derivation.
Thank you!
 A: Yes, it comes from antisymmetry of $\omega$.  Equating the terms involving the infinitesimal parameters $\omega$ on both sides of eq. 2.4.7 gives (hopefully my indices are all right here!)
\begin{align}
  \frac{1}{2}&\omega_{\rho\sigma}U(\Lambda, a) J^{\rho\sigma} U^{-1}(\Lambda, a) \\
  &= \frac{1}{2}(\Lambda\omega\Lambda^{-1})_{\mu\nu}J^{\mu\nu} + (\Lambda\omega\Lambda^{-1}a)_\mu P^\mu \\
  &= \frac{1}{2}\Lambda_\mu^{\phantom\mu\rho}\omega_{\rho\sigma}
  (\Lambda^{-1})^\sigma_{\phantom\sigma\nu} J^{\mu\nu} + \Lambda_\mu^{\phantom\mu\rho}\omega_{\rho\sigma}(\Lambda^{-1})^\sigma_{\phantom\sigma\nu}a^\nu P^\mu \\
  &= \frac{1}{2}\Lambda_\mu^{\phantom\mu\rho}\omega_{\rho\sigma}
  (\Lambda^{-1})^\sigma_{\phantom\sigma\nu} J^{\mu\nu} + \frac{1}{2}\Lambda_\mu^{\phantom\mu\rho}(\omega_{\rho\sigma}-\omega_{\sigma\rho})(\Lambda^{-1})^\sigma_{\phantom\sigma\nu}a^\nu P^\mu \\
  &= \frac{1}{2}\Lambda_\mu^{\phantom\mu\rho}\omega_{\rho\sigma}
  (\Lambda^{-1})^\sigma_{\phantom\sigma\nu} J^{\mu\nu} + \frac{1}{2}\Lambda_\mu^{\phantom\mu\rho}\omega_{\rho\sigma}(\Lambda^{-1})^\sigma_{\phantom\sigma\nu}a^\nu P^\mu -\frac{1}{2}\Lambda_\mu^{\phantom\mu\sigma}\omega_{\rho\sigma}(\Lambda^{-1})^\rho_{\phantom\sigma\nu}a^\nu P^\mu
\end{align}
now we identify coefficients of $\omega_{\rho\sigma}$ (and thus eliminate it) to obtain
\begin{align}
  U(\Lambda, a) J^{\rho\sigma} U^{-1}(\Lambda, a)
  &= \Lambda_\mu^{\phantom\mu\rho}
  (\Lambda^{-1})^\sigma_{\phantom\sigma\nu} J^{\mu\nu} + \Lambda_\mu^{\phantom\mu\rho}(\Lambda^{-1})^\sigma_{\phantom\sigma\nu}a^\nu P^\mu -\Lambda_\mu^{\phantom\mu\sigma}(\Lambda^{-1})^\rho_{\phantom\sigma\nu}a^\nu P^\mu \\
  &= \Lambda_\mu^{\phantom\mu\rho}
  \Lambda_\nu^{\phantom\nu\sigma} J^{\mu\nu} + \Lambda_\mu^{\phantom\mu\rho}\Lambda_\nu^{\phantom\nu\sigma}a^\nu P^\mu -\Lambda_\mu^{\phantom\mu\rho}\Lambda_\nu^{\phantom\nu\sigma}a^\mu P^\nu \\
  &= \Lambda_\mu^{\phantom\mu\rho}
  \Lambda_\nu^{\phantom\nu\sigma} (J^{\mu\nu} + a^\nu P^\mu -a^\mu P^\nu).
\end{align}
