  Let G be the split special orthogonal group of degree 2n+1 over a field F of
char F \ne 2. Then we describe G-orbits on the triple flag varieties G/P\times
G/P\times G/P and G/P\times G/P\times G/B with respect to the diagonal action
of G where P is a maximal parabolic subgroup of G of the shape (n,1,n) and B is
a Borel subgroup. As by-products, we also describe GL_n-orbits on G/B,
Q_{2n}-orbits on the full flag variety of GL_{2n} where Q_{2n} is the
fixed-point subgroup in Sp_{2n} of a nonzero vector in F^{2n} and 1\times
Sp_{2n}-orbits on the full flag variety of GL_{2n+1}. In the same way, we can
also solve the same problem for SO_{2n} where the maximal parabolic subgroup P
is of the shape (n,n).
