Lagrangian pairs and Lagrangian orthogonal matroids

Represented Coxeter matroids of types Cn and Dn, that is, symplectic and orthogonal matroids arising from totally isotropic subspaces of symplectic or (even-dimensional) orthogonal spaces, may also be represented in buildings of type Cn and Dn, respectively (see [4, Chapter 7]). Indeed, the particular buildings involved are those arising from the flags or oriflammes, respectively, of totally isotropic subspaces. There are also buildings of type Bn arising from flags of totally isotropic subspaces in odd-dimensional orthogonal space. Coxeter matroids of type Bn are the same as those of type Cn (since they depend only upon the directions of roots, not the length of roots—see the Gelfand-Serganova Theorem below). However, buildings of type Bn are distinct from those of the other types. Thus the question arises ∗Supported by EPSRC Grant GR/R53593.