How to Simplify Commutators Using Levi-Civita Symbol?

[KostasV]

Homework Statement

The problem statement can be seen in the picture i uploaded.

Homework Equations

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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 don't 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)

 

[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.

 

[KostasV]

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.

I can't see why this is wrong ... :/

 

[KostasV]

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.

Moreover , if i use , let's 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 ...

 

[KostasV]

Ok i think i understand why i can't 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)

 

[TSny]

Yes. That looks very good.

 

[KostasV]

Yes. That looks very good.

Thank you very much for your help :D