  This is a continuation of our "Lecture on Kac--Moody Lie algebras of the
arithmetic type" \cite{25}.
  We consider hyperbolic (i.e. signature $(n,1)$) integral symmetric bilinear
form $S:M\times M \to {\Bbb Z}$ (i.e. hyperbolic lattice), reflection group
$W\subset W(S)$, fundamental polyhedron $\Cal M$ of $W$ and an acceptable
(corresponding to twisting coefficients) set $P({\Cal M})\subset M$ of vectors
orthogonal to faces of $\Cal M$ (simple roots). One can construct the
corresponding Lorentzian Kac--Moody Lie algebra ${\goth g}={\goth
g}^{\prime\prime}(A(S,W,P({\Cal M})))$ which is graded by $M$.
  We show that $\goth g$ has good behavior of imaginary roots, its denominator
formula is defined in a natural domain and has good automorphic properties if
and only if $\goth g$ has so called {\it restricted arithmetic type}. We show
that every finitely generated (i.e. $P({\Cal M})$ is finite) algebra ${\goth
g}^{\prime\prime}(A(S,W_1,P({\Cal M}_1)))$ may be embedded to ${\goth
g}^{\prime\prime}(A(S,W,P({\Cal M})))$ of the restricted arithmetic type. Thus,
Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type is a
natural class to study.
  Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type have the
best automorphic properties for the denominator function if they have {\it a
lattice Weyl vector $\rho$}. Lorentzian Kac--Moody Lie algebras of the
restricted arithmetic type with generalized lattice Weyl vector $\rho$ are
called {\it elliptic}
