  Represented Coxeter matroids of types $C_n$ and $D_n$, 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 $C_n$ and $D_n$, respectively. Indeed, the particular buildings
involved are those arising from the flags or oriflammes, respectively, of
totally isotropic subspaces. There are also buildings of type $B_n$ arising
from flags of totally isotropic subspaces in odd-dimensional orthogonal space.
Coxeter matroids of type $B_n$ are the same as those of type $C_n$ (since they
depend only upon the reflection group, not the root system). However, buildings
of type $B_n$ are distinct from those of the other types. The matroids
representable in odd dimensional orthogonal space (and therefore in the
building of type $B_n$) turn out to be a special case of symplectic (flag)
matroids, those whose top component, or Lagrangian matroid, is a union of two
Lagrangian orthogonal matroids. These two matroids are called a Lagrangian
pair, and they are the combinatorial manifestation of the ``fork'' at the top
of an oriflamme (or of the fork at the end of the Coxeter diagram of $D_n$).
  Here we give a number of equivalent characterizations of Lagrangian pairs,
and prove some rather strong properties of them.
