Non-commuting operators

[S.G.]

How do you prove the following identity for non-commuting operators A and B?

[[[A,B],B]A]=[B,[A,[A,B]]]

 

[Edgardo]

Hello S.G.,

you could first use the abbreviation C:= [A,B], so the equation
looks better:

[[[A,B],B]A]=[B,[A,[A,B]]]
<=>
[[C,B],A] = [B,[A,C]]

Then use the definition of the commutator [Y,Z] = YZ-ZY
and in the end, instead of C, use [A,B] again.