  As part of his classification of regular semisimple subalgebras of semisimple
Lie algebras, Dynkin introduced the notion of a $\pi$-system. This is a subset
of the roots such that pairwise differences of its elements are not roots.
These arise as simple systems of regular semisimple subalgebras. Morita and
Naito generalized this notion to all symmetrizable Kac-Moody algebras. In this
work, we systematically develop the theory of $\pi$-systems of symmetrizable
Kac-Moody algebras and establish their fundamental properties. We study the
orbits of the Weyl group on $\pi$-systems, and completely determine the number
of orbits in many cases of interest in physics. In particular, we show that
there is a unique $\pi$-system of type $HA_1^{(1)}$ (the Feingold-Frenkel
algebra) in $E_{10}$ (the rank 10 hyperbolic algebra) up to Weyl group action
and negation.
