Let R be any ring (with or without 1). For any r,s \in R define [r,s]=rs-sr. Suppose [[x,y],z]=0, for all x,y,z \in R. Prove that: [a,b][c,d] = [c,a][b,d], for all a,b,c,d \in R.


Suggestion:

Spoiler:
0=[[a,bc],d]= \cdots