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).