Examples to Birkhoff's quasigroup axioms

The equational variety of quasigroups is defined by six identities, called Birkhoff's identities. It is known, that only four of them suffice to define the variety; actually, there are nine different combinations of four Birkhoff's identities defining quasigroups, other four combinations define larger varieties and it was open whether the remaining two cases define quasigroups or larger classes. We solve the question here constructing examples of algebras that are not quasigroups and satisfy the open cases of Birkhoff's identities.