At the end of this page: http://dogschool.tripod.com/housekeeping.html
It proves the left sided axioms using the right sided axioms. It then asks you at then end to prove the two-sided axioms using a right-sided identity axiom and a left-sided inverse axiom.
If we know only:
$e • a = a$
And
$a • a^{-1} = e$
How do we prove:
$a • e = a$
And
$a^{-1} • a = e$