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We give a criterion for a Dynkin diagram, equivalently a generalized Cartanmatrix, to be symmetrizable. This criterion is easily checked on the Dynkindiagram. We obtain a simple proof that the maximal rank of a Dynkin diagram ofcompact hyperbolic type is 5, while the maximal rank of a symmetrizable Dynkindiagram of compact hyperbolic type is 4. Building on earlier classificationresults of Kac, Kobayashi-Morita, Li and Sa\c{c}lio\~{g}lu, we present the 238hyperbolic Dynkin diagrams in ranks 3-10, 142 of which are symmetrizable. Foreach symmetrizable hyperbolic generalized Cartan matrix, we give asymmetrization and hence the distinct lengths of real roots in thecorresponding root system. For each such hyperbolic root system we determinethe disjoint orbits of the action of the Weyl group on real roots. It followsthat the maximal number of disjoint Weyl group orbits on real roots in ahyperbolic root system is 4.

Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits