  Let us fix a complex simple Lie algebra and its non-compact real form. This
paper focuses on non-zero adjoint nilpotent orbits in the complex simple Lie
algebra meeting the real form. We show that the poset consisting of such
nilpotent orbits equipped with the closure ordering has the minimum.
Furthermore, we determine such the minimum orbit in terms of the
Dynkin--Kostant classification even in the cases where the orbit does not
coincide with the minimal nilpotent orbit in the complex simple Lie algebra. We
also prove that the intersection of the orbit and the real form is the union of
all minimal nilpotent orbits in the real form.
