## Algebra, Problems For Fun (34)

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