Maximal Abelian Sets of Roots

In this work the author lets $\Phi$ be an irreducible root system, with Coxeter group $W$. He considers subsets of $\Phi$ which are abelian, meaning that no two roots in the set have sum in $\Phi \cup \{ 0 \}$. He classifies all maximal abelian sets up to the action of $W$: for each $W$-orbit of maximal abelian sets we provide an explicit representative $X$.