  In this paper, we analyse the Einstein and Einstein-Maxwell billiards for all
spatially homogeneous cosmological models corresponding to 3 and 4 dimensional
real unimodular Lie algebras and provide the list of those models which are
chaotic in the Belinskii, Khalatnikov and Lifschitz (BKL) limit. Through the
billiard picture, we confirm that, in D=5 spacetime dimensions, chaos is
present if off-diagonal metric elements are kept: the finite volume billiards
can be identified with the fundamental Weyl chambers of hyperbolic Kac-Moody
algebras. The most generic cases bring in the same algebras as in the
inhomogeneous case, but other algebras appear through special initial
conditions.
