The algebras \({\mathcal A}\) for which the subalgebra lattice Sub \({\mathcal A}\times {\mathcal A}\) is a modular lattice are studied. It is proved that if \({\mathcal A}\) is supposed to be, in addition, an idempotent algebra, then it is the trivial algebra with the exception of one two-element algebra. Another theorem states that if Sub \({\mathcal A}^ 4\) is modular, then \({\mathcal A}\) has to satisfy the term condition, i.e. it is Abelian. The paper contains also a characterization of varieties with distributive subalgebra lattices.