  The hyperbolic (and more generally, Lorentzian) Kac-Moody (KM) Lie algebras
$\cA$ of rank $r+2 > 2$ are shown to have a rich structure of indefinite KM
subalgebras which can be described by specifying a subset of positive real
roots of $\cA$ such that the difference of any two is not a root of $\cA$.
Taking these as the simple roots of the subalgebra gives a Cartan matrix,
generators and relations for the subalgebra. Applying this to the canonical
example of a rank 3 hyperbolic KM algebra, $\cF$, we find that $\cF$ contains
all of the simply laced rank 2 hyperbolics, as well as an infinite series of
indefinite KM subalgebras of rank 3. It is shown that $\cA$ also contains
Borcherds algebras, obtained by taking all of the root spaces of $\cA$ whose
roots are in a hyperplane (or any proper subspace). This applies as well to the
case of rank 2 hyperbolics, where the Borcherds algebras have all their roots
on a line, giving the simplest possible examples.
