Abstract
We give a criterion for a Dynkin diagram, equivalently a generalized Cartan
matrix, to be symmetrizable. This criterion is easily checked on the Dynkin
diagram. We obtain a simple proof that the maximal rank of a Dynkin diagram of
compact hyperbolic type is 5, while the maximal rank of a symmetrizable Dynkin
diagram of compact hyperbolic type is 4. Building on earlier classification
results of Kac, Kobayashi-Morita, Li and Saclio\~glu, we present the 238
hyperbolic Dynkin diagrams in ranks 3-10, 142 of which are symmetrizable. For
each symmetrizable hyperbolic generalized Cartan matrix, we give a
symmetrization and hence the distinct lengths of real roots in the
corresponding root system. For each such hyperbolic root system we determine
the disjoint orbits of the action of the Weyl group on real roots. It follows
that the maximal number of disjoint Weyl group orbits on real roots in a
hyperbolic root system is 4.
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