Can someone help me prove these commutation relations?
$$[M_{ij},M_{kl}]=\delta_{jk}M_{il}+\delta_{il}M_{jk}-\delta_{ik}M_{jl}-\delta_{jl}M_{ik}$$
where $i, j, k, l$ run from $1$ to $3$.
And:
$$[M_{\mu\nu},M_{\lambda\rho}]=\eta_{\mu\rho}M_{\nu\lambda}+\eta_{\nu\lambda}M_{\mu\rho}-\eta_{\mu\lambda}M_{\nu\rho}-\eta_{\nu\rho}M_{\mu\lambda}~?$$
I know that $\partial_{i}x_{j}$ gives us $\eta_{ij}$, but why am I getting Kronecker delta in the first expression? Is it because of the Euclidean space? I read somewhere that $\partial_{i}x^{j}$ gives us $\delta_{i}^{j}$, then why am I getting Kronecker delta in the above expression instead of $\eta$?
