  We classify regular subalgebras of affine Kac-Moody algebras in terms of
their root systems. In the process, we establish that a root system of a
subalgebra is always an intersection of the root system of the algebra with a
sublattice of its root lattice.
  We also discuss applications to investigations of regular subalgebras of
hyperbolic Kac-Moody algebras and conformally invariant subalgebras of affine
Kac-Moody algebras.
