  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.
