A Geometric Characterization of the Hjelmslev–Moufang Planes

Abstract Hjelmslev–Moufang (HM) planes are point-line geometries related to the exceptional algebraic groups of type $\mathsf{E}_6$. More generally, spherical Tits buildings—Lie incidence geometries—are prominent examples parapolar spaces: axiomatically defined consisting points, lines and symplecta (structures isomorphic polar spaces). In this paper we classify spaces with a similar behaviour as HM planes, in sense that their never have non-empty intersection. Under standard assumptions, obtain only such exactly given by close relatives (arising from taking certain restrictions). On one hand, work complements approach using Jordan algebras due Faulkner his book ‘The Role Nonassociative Algebra Projective Geometry’, published American Mathematical Society 2014; on other it provides new tool for classification characterization problems general theory spaces.