Decomposition of phase space and classification of Heisenberg groups

Is every locally compact abelian group which admits a Heisenberg central extension isomorphic to the product of a locally compact abelian group and its Pontryagin dual? An affirmative answer is obtained for all the commonly occurring types of abelian groups having Heisenberg central extensions, including Lie groups and certain finite Cartesian products of local fields and adeles. Furthermore, for these types of groups, it is found that the isomorphism class of the abelian group determines the Heisenberg group up to isomorphism, thereby providing a classification of such Heisenberg groups.