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


Suggestion:

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