Orthomodular lattices from 3-dimensional quadratic spaces

If E is a vector space over a field K, then any regular symmetric bilinear form phi on E induces a polarity M --> M-perpendicular to on the lattice of all subspaces of E. In the particular case where E is 3-dimensional, the set of all subspaces M of E such that both M and M-perpendicular to are not N-subspaces (which, in most cases, is equivalent to saying that M is nonisotropic), ordered by inclusion and endowed with the restriction of the above polarity, is an orthomodular lattice T(E, phi). We show that if K' is a proper subfield of K, with K'not equalF(2), and E' a 3-dimensional K'-subspace of E such that the restriction of phi to E' x E' is, up to multiplicative constant, a bilineax form phi' on the K'-space E', then T(E', phi') is isomorphic to an irreducible 3-homogeneous proper subalgebra of T(E, phi). Our main result is a structure theorem stating that, when K is not of characteristic 3, the converse is true, i.e., any irreducible 3-homogeneous proper subalgebra of T(E, phi) is of this form. As a corollary, we construct infinitely many finite orthomodular lattices which are minimal in the sense that all their proper subalgebras are modular. In fact, this last result was our initial aim in this paper.