Orthomodular lattices of subspaces obtained from quadratic forms

Being given a field K of characteristic different from 2 and 3, a 3-dimensional vector space E over K, and a nonsingular symmetric bilinear form. over E, we define a structure of orthomodular lattice T (E,phi) on the set of all nonisotropic subspaces of E. We give a structure Theorem about the irreducible and 3-homogeneous subalgebras of T (E,phi). In particular, these subalgebras are all of the form T (E',phi') where E' is a 3-dimensional subspace of E, if E is regarded as a vector space over a subfield K' of K, and phi' is induced by phi. This structure Theorem allows us to achieve an old project, concerning minimal orthomodular lattices (an orthomodular lattice L is called minimal if it is nonmodular and if all its proper subalgebras are either modular, or isomorphic to L).