Combining the associative and commutative property into one

Can anybody do better than this?

Say I define a specific type of algebraic structure (I guess, a commutative monoid) with the following axioms:

1) There exists an identity element in the structure (G) e such that for all a in G, ae = a.

2) For all a, b, and c in G, (ab)c = b(ac)

And from only these we can deduce the commutative and associative axioms!

ab

= (ab)

= (ab)e (identity axiom)

= b(ae) (axiom #2)

= b(a) (identity axiom)

= ba

And thus ab = ba. And now for the associative property,

(ab)c

= (ba)c (commutative property)

= a(bc) (axiom #2)

Is there any statement that would deduce both the commutative and associative properties without the need of an identity element?