Double products and hypersymplectic structures on $R^{4n}$

In this paper we give a procedure to construct hypersymplectic structures on $R^{4n}$ beginning with affine-symplectic data on $R^{2n}$. These structures are shown to be invariant by a 3-step nilpotent double Lie group and the resulting metrics are complete and not necessarily flat. Explicit examples of this construction are exhibited.