  Let G be a reductive algebraic group in classical types A, B, D and e be an
element of its Lie algebra with Z its centraliser in G for the adjoint action.
We suppose that e identifies with an nilpotent matrix of order two, which
guarantees the number of Z-orbits in the flag variety of G is finite. For types
B, D in characteristic two, we also suppose the image of e is totally
isotropic. We show that any closure Y of such orbit is normal. We also prove
that Y is Cohen-Macaulay with rational singularities provided that the base
field is of characteristic zero, and that Cohen-Macaulayness remains in any
characteristic for type A. We exhibit a birational, rational morphism onto Y
involving Schubert varieties. Our work generalizes a result by N. Perrin and E.
Smirnov on Springer fibers ([PS12]).
