  A complete description of the coadjoint orbits for A_{n-1}^{+}, the nilpotent
Lie algebra of n-by-n strictly upper triangular matrices, has not yet been
obtained, though there has been steady progress on it ever since the orbit
method was devised.
  We apply methods developed by Andre' to find defining equations for the
elementary coadjoint orbits for the maximal nilpotent Lie subalgebras of the
orthogonal
  Lie algebras, and we also determine all the possible dimensions of coadjoint
orbits in the case of A_{n-1}^{+}.
