Abelian ideals of a Sorel subalgebra and root systems

Let g be a simple Lie algebra and 216 degrees the poset of non-trivial abelian ideals of a fixed Borel subalgebra of g. In [8], we constructed a partition 216 degrees = Li-mu 216 mu parameterised by the long positive roots of g and studied the subposets 216(mu). In this note, we show that this partition is compatible with intersections, relate it to the Kostant Peterson parameterisation and to the centralisers of abelian ideals. We also prove that the poset of positive roots of g is a join-semilattice.