commuting, non-commuting, anti-commuting

We have



[A,B] equals zero, commuting



[A,B] not equals zero, not commuting



[A,B] = - [B,A] , anti-commuting





So then we can say



[X,P] anticommutes, since [X,P]= -[P,X] , and



[X,P] does not commute, since [X,P] = ih



I find that confusing. Is there something I missed? (Is this on the one hand math language for the Lie algebra, which needs to be anti-commuting, and on the other hand physics language for commuting and non-commuting observables?)



Also, for femions there is the anti-commuting relations {A,B}. Here A,B anticommute if {A,B} is zero. But X and P for bosons anticommute, why are we here not using the anticommutator.



Can someone unconfuse me?

thanks