We show there is a class of symplectic Lie algebra representations over anyfield of characteristic not 2 or 3 that have many of the exceptional algebraicand geometric properties of both symmetric three forms in two dimensions andalternating three forms in six dimensions. All nonzero orbits are coisotropicand the covariants satisfy relations generalising classical identities ofEisenstein and Mathews. The main algebraic result is that suitably genericelements of these representation spaces can be uniquely written as the sum oftwo elements of a naturally defined Lagrangian subvariety. We give universalexplicit formulae for the summands and show how they lead to the existence ofgeometric structure on appropriate subsets of the representation space. Overthe reals this structure reduces to either a conic, special pseudo-K\" ahlermetric or a conic, special para-K\" ahler metric.
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