An important class of Coxeter groups are the finite ones, like above. One can show that a
Coxeter group is finite if and only if the scalar product defined by Equation (3.28) on is
Euclidean . Finite Coxeter groups coincide with finite reflection groups in Euclidean space
(through hyperplanes that all contain the origin) and are discrete subgroups of . The
classification of finite Coxeter groups is known and is given in Table 1 for completeness. For finite
Coxeter groups, one has the important result that the Tits cone coincides with the entire space
Finite Coxeter groups.
"Spacelike Singularities and Hidden Symmetries of Gravity"
and Daniel Persson
and Philippe Spindel