A K\"{a}hler and quaternion-K\"{a}hler spacetime structure

When real Lorentzian spacetime is embedded into a manifold parametrized by higher division algebras (complex or quaternion with Hermitean metric) and the representation constraints of their symmetry groups are made compatible, a set of quantum numbers is generated that is evocative of those of the standard model of particle physics. This is taken here as a hint that in spacetime there is a pseudo-K\"{a}hler or pseudo-quaternion-K\"{a}hler structure, real spacetime being a submanifold that inherits the symmetry contraints of the larger ambient manifold.