I was trying to prove that: $$[P_\mu, J_{\rho \sigma}] = i(\eta_{\mu \sigma} P_\rho - \eta_{\mu \rho} P_\sigma) $$
$\textbf{Attempt}$
$$\begin{align} [P_\mu, J_{\rho \sigma}] = [P_\mu, x_\rho P_\sigma - x_\sigma P_\rho] \\ =[P_\mu, x_\rho P_\sigma] - [P_\mu, x_\sigma P_\rho] \\ =-i[x_\sigma , \partial_\mu]P_\rho+ i[x_\rho , \partial_\mu]P_\sigma \\ =-i(x_\sigma \partial_\mu - \partial_\mu x_\sigma) P_\rho + i(x_\rho \partial_\mu - \partial_\mu x_\rho)P_\sigma \end{align}$$
using $\partial_\mu x_\rho = \eta_{\mu \rho} $
$$[P_\mu, J_{\rho \sigma}] = -i(x_\sigma \partial_\mu - \eta_{\mu \sigma}) P_\rho + i(x_\rho \partial_\mu - \eta_{\mu \rho})P_\sigma \\ =i(\eta_{\mu \sigma} P_\rho - \eta_{\mu \rho}P_\sigma) $$
However in this post Commutation of position four-vector with spacetime derivatives they get:
\begin{align} [P_a,M_{bc}] =g_{ab}\partial_c - g_{ac}\partial_b \end{align}
That is the negative result of what I got, what am I doing wrong?
