A root system is a subset 
of a Euclidean space 
satisfying the conditions below. Let
denote the dot
product, and denote the reflection in the hyperplane 
by
 |
where
 |
The defining conditions on 
are:
1. 
is finite,
spans 
, and does not contain 0,
2. If 
, the reflection 
leaves 
invariant, and
3. If 
, then 
.
The Lie algebra roots of a finite-dimensional complex semisimple Lie algebra form a root
system in a real subspace of the dual
vector space to the Cartan subalgebra. In
this case, the reflections 
generate the Weyl group,
which is the subgroup of the symmetry
group of the root system generated by these reflections.
This subgroup can be proper. For example, the six vectors from the center to the
vertices of a regular hexagon
form a root system whose symmetry group has group order 12, whereas its Weyl
group has group order 6.
