  In this paper we extend the theory of oriented matroids to Lagrangian
orthogonal matroids and their representations, and give a completely natural
transformation from a representation of a classical oriented matroid to a
representation of the same oriented matroid considered as a Lagrangian
orthogonal matroid. Classical concepts of orientation and this extension may
both be thought of as stratifications of thin Schubert cells into unions of
connected components.
