Issues with representations of SU(2) and the su(2) algebra

Question

I'm currently having a somewhat hard time with a rather simple task.

This is my first contact with the formalism of Lie Groups. I'm studying the SU(2) and SO(3) groups and their algebra su(2) and trying to express su(2) in different forms.

I know the algebra is defined by this commutation relation between the generators $$[J_i,J_j] = i \epsilon_{ijk}J_k$$

but I cannot prove that this relation can be expressed in other forms if we use different J definitions, for example

i)$$[J^{ij},J^{lm}]=i(\delta^{il}J^{jm}-\delta^{jl}J^{im}-\delta^{im}J^{jl}+\delta^{jm}J^{il})$$ if we use $J^{ij} \equiv \epsilon_{ijk}J_k$,

ii)the form the algebra takes when we use the definition $J_i \equiv -i(x_j\partial_k-x_k\partial_j)$

I want to be able to prove that the algebra can be represented in these different forms, but every time I apply the J definitions to the commutator I ended up with expressions that don't lead to the correct forms.

I don't know if someone else had a tough time while studying this subject for the first time, but I appreciate any help. Thank you all.

Answer 1

For 2:

First, $$ (x_j \partial_k)(x_m \partial_n) = x_j (\partial_k x_m) \partial_n + (x_j x_m) (\partial_k \partial_n) = x_j \delta_{km} \partial_n + (x_j x_m) (\partial_k \partial_n) . $$

Therefore, $$\begin{align} [x_j \partial_k, x_m \partial_n] &= (x_j \partial_k)(x_m \partial_n) - (x_m \partial_n)(x_j \partial_k) \\ &= (x_j \delta_{km} \partial_n + (x_j x_m) (\partial_k \partial_n)) - x_m \delta_{jn} \partial_k + (x_m x_j) (\partial_n \partial_k) \\ &= \delta_{km} x_j \partial_n - \delta_{jn} x_m \partial_k \\ \end{align}$$ and $$\begin{align} [x_j \partial_k - x_k \partial_j, x_m \partial_n - x_n \partial_m] & = [x_j \partial_k, x_m \partial_n] - [x_j \partial_k, x_n \partial_m] - [x_k \partial_j, x_m \partial_n] + [x_k \partial_j, x_n \partial_m] \\& = (\delta_{km} x_j \partial_n - \delta_{jn} x_m \partial_k) - (\delta_{kn} x_j \partial_m - \delta_{jm} x_n \partial_k) \\& - (\delta_{jm} x_k \partial_n - \delta_{kn} x_m \partial_j) + (\delta_{jn} x_k \partial_m - \delta_{km} x_n \partial_j) \\& = \delta_{km} (x_j \partial_n - x_n \partial_j) - \delta_{jn} (x_m \partial_k - x_k \partial_m) \\& - \delta_{kn} (x_j \partial_m - x_m \partial_j) + \delta_{jm} (x_n \partial_k - x_k \partial_n) . \end{align}$$