I'm working through problems in the book Einstein Gravity in a Nutshell by Zee, and I'm stuck on one of the harder problems. The problem is
Calculate $[J_{(mn)}, J_{(pq)}]$.
We are given that $[J_{(mn)}, J_{(pq)}] = i(\delta_{mp}J_{(np)} + \delta_{nq}J_{(mp)} - \delta_{np}J_{(mq)} - \delta_{mq}J_{(np)})$, and that $J^{ij}_{(mn)} = -i(\delta^{mi}\delta^{nj} - \delta^{mj}\delta^{ni})$.
I started off writing the commutator definition: $[J_{(mn)}, J_{(pq)}] = J_{(mn)}J_{(pq)} - J_{(pq)}J_{(mn)}$.
I'm rather new to indices and I keep getting stuck on how to write the above equation in index notation. I'm also a little confused on how to move between the $\delta_{ij}$ and $\delta^{ij}$ in the problem. Is there a trick I'm missing or is this just a problem involving a lot of algebra?
I see that there is another question here that is very similar, although I think my question is more about doing the algebra with the indices. This might be the wrong place to ask such a question, though.
