  We consider a natural generalisation of the class of hyperbolic Kac-Moody
algebras. We describe in detail the conditions under which these algebras are
Lorentzian. We also construct their fundamental weights, and analyse whether
they possess a real principal so(1,2) subalgebra. Our class of algebras include
the Lorentzian Kac-Moody algebras that have recently been proposed as
symmetries of M-theory and the closed bosonic string.
