[math-fun] abelian VS commutative & capitalization

Is there any rule or convention about "abelian" VS "commutative"? For example, I see "abelian group" but "commutative monoid"? With just a little grousing I can accept the non-descriptive but honorific "abelian" as the preferred synonym, but then why not use it consistently? Also, what about capitalization? I've seen it both ways, but personally prefer lower-case on the grounds that once a proper name is *modified* into an adjective (as opposed to used unchanged, as in "Erdos number") the word no longer "properly" refers to the person (similarly for "riemannian"). And then what about "the" "gudermannian" and "pfaffian" which are adjectival forms of names now apparently used as either nouns or adjectives?