Operator B has a commutator [A,B]

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I'm working on a question where the operator B has a commutator [A,B] which is not equal to AB-BA, and instead has a squared term in it. Maybe I just don't understand commutators but what does this mean/how do I use the commutator (to find eigenvectors of A)?
 
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Well, you have the right understanding as far as I can tell. Can you post the example in question? You haven't been too specific with your question.
 

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