Symmetric and skew-symmetric complex structures

On a complex manifold (M,J), we interpret complex symplectic and pseudo-Kähler structures as symplectic forms with respect to which J is, respectively, symmetric and skew-symmetric. We classify complex symplectic structures on 4-dimensional Lie algebras. We develop a method for constructing hypersymplectic structures from the above data. This allows us to obtain two families of hypersymplectic structures on a 4-step nilmanifold.