On adjoint orbits in nilpotent ideals of a Borel subalgebra

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 $B$. We prove that $\mathfrak{m}^{\bullet}$ contains a unique closed $B$-orbit which is the orbit of a nilpotent element whose support is the set of minimal roots associated to the root space decomposition of $\mathfrak{m}$.