On ${\pi}$-systems of symmetrizable Kac-Moody algebras

Given a symmetrizable Kac-Moody algebra $\mathfrack{g}$, we study its $\pi$-systems, which are subsets of real roots, the pairwise differences of whose elements are not roots. Such systems arise as simple systems of regular subalgebras of $\mathfrack{g}$, and were originally studied by Dynkin, Morita and Naito. We show that the binary relation introduced by Morita defines a partial order on the set of $\mathfrack{g}$ of finite, untwisted affine or hyperbolic type. We also formulate general principles for constructing $\pi$-systems as well as for finding forbidden diagrams that cannot occur as Dynkin diagrams of $\pi$-systems of a given $\mathfrack{g}$. Among other applications, we use this to determine the set of maximal hyperbolic Dynkin diagrams in ranks $3$-$10$ relative to the Morita partial order.