Maximal Abelian Sets of Roots

Book excerpt: 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 (i.e., abelian sets properly contained in no other) up to the action of W: for each W-orbit of maximal abelian sets we provide an explicit representative X, identify the (setwise) stabilizer W_X of X in W, and decompose X into W_X-orbits. Abelian sets of roots are closely related to abelian unipotent subgroups of simple algebraic groups, and thus to abelia.