Derivation of angular momentum commutator relations

Question

I'm trying to understand the derivation of the angular momentum commutator relations. How is

$$[zp_y, zp_x] ~=~ 0?$$

How is

$$[yp_z, zp_x] ~=~ y[p_z, z]p_x?$$

Answer 1

Remember that

$$[AB,C] = A[B,C] + [A,C]B$$

With this, the first identity follows directly from the canonical commutation relation

$$[x_i,p_j] = i \hbar \delta_{ij}$$

along with

$$[x_i,x_j] = [p_i,p_j] = 0$$

For the second, the same relations tell us that $y$ and $p_x$ commute with everything else in sight, and so can be treated as C-numbers.