  Is every locally compact abelian group which admits a symplectic self-duality
isomorphic to the product of a locally compact abelian group and its Pontryagin
dual? Several sufficient conditions, covering all the typical applications are
found. Counterexamples are produced by studying a seemingly unrelated question
about the structure of maximal isotropic subgroups of finite abelian groups
with symplectic self-duality (where the original question always has an
affirmative answer).
