Abelian ideals of a Borel subalgebra and long positive roots

Let b be a Borel subalgebra of a simple Lie algebra g and let Ab denote the set of all Abelian ideals of b. We consider Ab as poset with respect to inclusion, the zero ideal being the unique minimal element of Ab. We show that there is a natural mapping τ from the set of all nontrivial Abelian ideals to the set of long positive roots. Each fibre of τ is a poset in its own right, and it is proved that this fibre has a unique maximal and a unique minimal element. If I is a maximal Abelian ideal, then we show that τ(I) is a (long) simple root. As a consequence, we deduce that restricting τ to the set of maximal Abelian ideals yields a one-to-one correspondence between the maximal Abelian ideals and the long simple roots. An explicit description of the minimal and the maximal ideal corresponding to a long root is given.