Lagrangian pairs and Lagrangian orthogonal matroids

Lagrangian orthogonal matroids are the best-behaved Coxeter matroids, except for ordinary matroids, and indeed they include ordinary matroids as a special case. Represented Lagrangian orthogonal matroids arise from n-dimensional totally orthogonal subspaces of 2n-dimensional orthogonal space, and if we take the unique pair of such subspaces which contain a given n−1-dimensional totally isotropic subspace, we get the prototype of a Lagrangian pair of Lagrangian orthogonal matroids. In this paper we define Lagrangian pairs, and give a number of characterizations of them and prove several properties of them. We also use them to prove a new characterization, within ordinary matroid theory, of an elementary quotient of ordinary matroids.