Projective spaces of second order

Grassmannians of higher order appeared for the first time in a paper of A. Szybiak ([3]) in the context of the Cartan method of moving frame. In the present paper we consider a special case of higher order Grassmannian—the projective space of second order. We introduce the projective group of second order acting on this space, derive its Maurer-Cartan equations and show that our generalized projective space is a homogeneous space of this group.