The class of uniquely complemented lattices is not a subvariety of the class of complemented lattices. On the other hand, there is a well-known class
of uniquely complemented lattices that is a subvariety
of the variety of complemented lattices, namely the class of Boolean
algebras. They form a variety because they are the
distributive complemented lattices, and one can prove that any distributive complemented
lattice is uniquely complemented.