Generalized quadrangles and cubic forms

A Jordan pair is constructed from a pair of cubic forms satisfying the adjo int identities. Given some parameters and an incidence structure S having t hree points on each line and no more than one line through two points, a pa ir of cubic forms are constructed. These forms satisfy the adjoint identiti es if and only if S is either a star or a generalized quadrangle and the pa rameters are precisely determined.