Basic commutator of angular momentum

[catsarebad]

Could someone explain to me how the author goes from 2nd to 3rd step

I think the intermediate step between 2 and 3 is basically to split up the commutator as

[y p_z, z p_x] - [y p_z,x p_z] - [z p_y,z p_x] + [z p_y, x p_z]

2nd term = 0
3rd term = 0

so leftover is
[L_x, L_y] = [y p_z, z p_x] + [z p_y, x p_z]

but how does this turn into what he has on 3rd step?

 

[strangerep]

[...]
so leftover is
[L_x, L_y] = [y p_z, z p_x] + [z p_y, x p_z]

but how does this turn into what he has on 3rd step?

Multiple applications of the Leibniz product rule: $[AB,C] = A[B,C] + [A,C]B$

 

[catsarebad]

really? I thought it would be much simpler than that. I thought i was missing a trivial trick.

 

[strangerep]

really? I thought it would be much simpler than that. I thought i was missing a trivial trick.

Once you become proficient with the Leibniz rule, you'll be able to skip steps. E.g., the $y$ in the 1st commutator commutes with $zp_x$, so it can just be taken out the front, and so on. You could call that a "trivial" trick, but it's wise to carefully practice the Leibniz rule a few times initially, since it's essential when simplifying more difficult commutators.