Minimal Nilpotent Orbits of type D and E

We first show the closure of the minimal nilpotent adjoint orbit Omin^{D_n} in so_{2n} is isomorphic to the affinization of T^*(SL_{n-1}/[P,P]) where P is the parabolic subgroup P_{(1,1,n-3)} of SL_{n-1}(C). Then we prove that the closure of the minimal nilpotent adjoint orbit Omin^{E_6} of the complex simple Lie algebra E_6 is isomorphic to the affinization of T^*(SL_4/P^u) where P^u is the unipotent radical of the parabolic subgroup P_{(2,2)} of SL_4(\C). In the end we will formulate a similar result for type E_7.