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.

  • $\begingroup$ Possible duplicate: physics.stackexchange.com/q/28535/2451 $\endgroup$ – Qmechanic Jun 9 '19 at 19:00
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    $\begingroup$ For the true nerd: it should be $so(n)$ (lower case). It is usual to denote the algebra by lower case and the group by upper case. $\endgroup$ – ZeroTheHero Jun 9 '19 at 21:41
  • $\begingroup$ @ZeroTheHero oh okay, I think I learned that somewhere but very quickly forgot it. My bad. $\endgroup$ – King Nerd the Third Jun 9 '19 at 21:42