Let \(\mathfrak {m}\) be a nilpotent ideal in the Borel subalgebra \(\mathfrak {b}\) of a complex finite-dimensional semisimple Lie algebra, and \(\mathfrak {m}^{\bullet }\) the subset of (ad-)nilpotent elements in \(\mathfrak {b}\) such that \(\mathfrak {m}\) is the minimal ideal containing them. This set is stable under the adjoint action of the corresponding Borel subgroup \(\textbf{B}\) . We prove that \(\mathfrak {m}^{\bullet }\) contains a unique closed \(\textbf{B}\) -orbit which is the orbit of a nilpotent element whose support is the set of minimal roots associated to the root space decomposition of \(\mathfrak {m}\) .
