Minimal inversion complete sets and maximal abelian ideals

Let $\mathfrak g$ be a simple Lie algebra, $\mathfrak b$ a fixed Borel subalgebra, and $W$ the Weyl group of $\mathfrak g$. In this note, we study a relationship between the maximal abelian ideals of $\mathfrak b$ and the minimal inversion complete sets of $W$. The latter have been recently defined by Malvenuto et al. (J. Algebra, 424 (2015), 330-356.)