Commutators and cevi levita symbol

1. Aug 23, 2015

KostasV

1. The problem statement, all variables and given/known data
The problem statement can be seen in the picture i uploaded.

2. Relevant equations
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3. The attempt at a solution
The attempt to the solution can be seen in the picture i uploaded.
I reached to the A and i dont know how to proceed to the solution that is given below. How does the minus and δkj disappear?
If i do the double summation on k and j I think that every term gets zero either because of εijk (levi-cevita) or δkj (kronecker)

2. Aug 23, 2015

TSny

You used the index j twice: once for the index on x and then again as a dummy summation index when writing out the angular momentum in terms of the levi-civita symbol.

3. Aug 23, 2015

KostasV

I cant see why this is wrong ... :/

4. Aug 23, 2015

KostasV

Moreover , if i use , lets say the index m on x (not on x that comes from angular momentum , yes on x that is alone) , then i still have the minus on ih bar ... My solutions say that it should not be there ...

5. Aug 23, 2015

KostasV

Ok i think i understand why i cant have the same index on these two .
Moreover i think i found how i get rid of this minus ...
I must use the fact that εijk=-εikj wright ?
Is now the solution correct ? (Uploaded photo)

6. Aug 23, 2015

TSny

Yes. That looks very good.

7. Aug 23, 2015

KostasV

Thank you very much for your help :D