  Let $\mathcal{D}$ be a Dynkin diagram and let $\Pi=\{\alpha_1,\dots
,\alpha_{\ell}\}$ be the simple roots of the corresponding Kac--Moody root
system. Let $\mathfrak{h}$ denote the Cartan subalgebra, let $W$ denote the
Weyl group and let $\Delta$ denote the set of all roots. The action of $W$ on
$\mathfrak{h}$, and hence on $\Delta$, is the discretization of the action of
the Kac--Moody algebra. Understanding the orbit structure of $W$ on $\Delta$ is
crucial for many physical applications. We show that for $i\neq j$, the simple
roots $\alpha_i$ and $\alpha_j$ are in the same $W$--orbit if and only if
vertices $i$ and $j$ in the Dynkin diagram corresponding to $\alpha_i$ and
$\alpha_j$ are connected by a path consisting only of single edges. We
introduce the notion of `the Cayley graph $\mathcal{P}$ of the Weyl group
action on real roots' whose connected components are in one-to-one
correspondence with the disjoint orbits of $W$. For a symmetric hyperbolic
generalized Cartan matrix $A$ of rank $\geq 4$ we prove that any 2 real roots
of the same length lie in the same $W$--orbit. We show that if the generalized
Cartan matrix $A$ contains zeros, then there are simple roots that are
stabilized by simple root reflections in $W$, that is, $W$ does not act simply
transitively on real roots. We give sufficient conditions in terms of the
generalized Cartan matrix $A$ (equivalently ${\mathcal D}$) for $W$ to
stabilize a real root. Using symmetry properties of the imaginary light cone in
the hyperbolic case, we deduce that the number of $W$--orbits on imaginary
roots on a hyperboloid of fixed radius is bounded above by the number of root
lattice points on the hyperboloid that intersect the closure of the fundamental
region for $W$.
