Abelian ideals of a Borel subalgebra and root systems

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Let g be a simple Lie algebra and Ab the poset of non-trivial abelian ideals of a fixed Borel subalgebra of g. In 2003 (IMRN, no.35, 1889--1913), we constructed a partition of Ab into the subposets Abμ, parameterised by the long positive roots of g, and established some properties of these subposets. In this note, we show that this partition is compatible with intersections, relate it to the Kostant-Peterson parameterisation of abelian ideals and to the centralisers of abelian ideals. We also prove that the poset of positive roots of g is a join-semilattice.