  Sometimes a hyperbolic Kac-Moody algebra admits an automorphic correction,
meaning a generalized Kac-Moody algebra with the same real simple roots and
whose denominator function has good automorphic properties; these for example
allow one to work out the root multiplicities. Gritsenko and Nikulin have
formalized this in their theory of Lorentzian Lie algebras and shown that the
real simple roots must satisfy certain conditions in order for the algebra to
admit an automorphic correction. We classify the hyperbolic root systems of
rank 3 that satisfy their conditions and have only finite many simple roots, or
equivalently a timelike Weyl vector. There are 994 of them, with as many as 24
simple roots. Patterns in the data suggest that some of the non-obvious cases
may be the richest.
