Ad-nilpotent ideals of a Borel subalgebra II

We provide an explicit bijection between the ad-nilpotent ideals of a Borel subalgebra of a simple Lie algebra and the orbits of under the Weyl group ( being the coroot lattice and h the Coxeter number of ). From this result we deduce in a uniform way a counting formula for the ad-nilpotent ideals.