# Commutators with powers? A 'show that' question

1. Oct 27, 2008

### Talvon

1. The problem statement, all variables and given/known data

The question is 'show that the commutator [AB,C]=A[B,C]+[A,C]B'

2. Relevant equations

I'm not sure, a search for a proof gave the names 'ring theory' and 'Leibniz algebra', but further searching hasn't provided a proof so far and it seems it is just accepted as a standard identity. Ring theory isn't on my syllabus and I haven't heard of it before today :|

3. The attempt at a solution

I tried applying the arbitrary commutators and I got ABC-CAB, and now I'm stuck lol as I can't arrange it to fit the above. Any help would be greatly appreciated, as I'm most likely going about it the wrong way, any identities for commutators with multiplied operators within them would be useful to know :P

Thanks :)

2. Oct 27, 2008

### Irid

Introduce a test function f:

[AB,C]f = ABCf - CABf = A([B,C]- CB)f + ([A,C]-AC)f

and work from there. Then just drop the test function f.

3. Oct 28, 2008

### Talvon

Thanks for the tip but I think I managed to do it without the test function (I think :P)

Could somebody please go over my workings to confirm if they are correct or not?

Expand:
[AB,C] = ABC - CAB

Sub in a factor of (ACB-ACB):
[AB,C] = ABC + (ACB - ACB) - CAB

Re-arrange:
(ABC - ACB) + (ACB - CAB)

Take out common factors:

A(BC - CB) + (AC - CA)B

Use commutator identities below:
A[B,C] = A(BC - CB)
[A,C]B = (AC - CA)B

Leading to A[B,C] + [A,C]B with any luck :P

Cheers :)