alt Hacker NewsCommutativity is a much weaker claim because one is totally before or after the other. e.g. AB may commute with C so ABC=CAB but it is not necessarily the case that this equals ACB. With asynchrony you are guaranteed ABC=ACB=CAB. (There may be an exisiting mathematical term for this but I don't know it)
You can prove three-term commutativity from two-term (I did it years ago, I think it looked something like this[1]), so the ordering doesn't matter.
[1] https://math.stackexchange.com/questions/785576/prove-the-co...