# Calculating the Commutation Relation of the Generators of $SO(n)$ [duplicate]

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.

## marked as duplicate by Qmechanic♦Jun 9 at 19:00

• 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. – ZeroTheHero Jun 9 at 21:41