  We identify the hyperbolic Kac Moody algebras for which there exists a
Lagrangian of gravity, dilatons and $p$-forms which produces a billiard that
can be identified with their fundamental Weyl chamber. Because of the
invariance of the billiard upon toroidal dimensional reduction, the list of
admissible algebras is determined by the existence of a Lagrangian in three
space-time dimensions, where a systematic analysis can be carried out since
only zero-forms are involved. We provide all highest dimensional parent
Lagrangians with their full spectrum of $p$-forms and dilaton couplings. We
confirm, in particular, that for the rank 10 hyperbolic algebra, $CE_{10} =
A_{15}^{(2)\wedge}$, also known as the dual of $B_8^{\wedge\wedge}$, the
maximally oxidized Lagrangian is 9 dimensional and involves besides gravity, 2
dilatons, a 2-form, a 1-form and a 0-form.
