Hyperbolic subalgebras of hyperbolic Kac-Moody algebras

We investigate regular hyperbolic subalgebras of hyperbolic Kac-Moody algebras via their Weyl groups. We classify all subgroups relations between Weyl groups of hyperbolic Kac-Moody algebras, and show that for every pair of a group and subgroup their exists at least one corresponding pair of algebra and subalgebra. We also present a finite algorithm classifying all regular hyperbolic subalgebras of hyperbolic Kac-Moody algebras.