In 1956, A. Selberg introduced weakly symmetric spaces in the framework of his development of the trace formula, and proved that in a weakly symmetric space, the algebra of all invariant (with respect to the full isometry group) differential operators is commutative [22]. In this paper, Selberg asked whether the converse holds. In the present work, we shall answer this question by presenting examples of commutative spaces which are not weakly symmetric. These examples arise in the quaternionic analogues to the Heisenberg group, endowed with certain special metrics (see Theorem 5 and the explicit realization after it).