whoa! is that true? i m not so sure. i think what you want to say here is:
[x,y]=[y,x] is a consequence of [x,x]=0, and if the field does not have characteristic 2, then [x,y]= [y,x] implies [x,x]=0, but not in fields with characteristic 2, so we drop that as an axiom.

Yes, that's essentially what I meant to say.