Let 
be a Euclidean space, 
be the dot product, and denote the reflection in
the hyperplane 
by
 |
where
 |
Then a subset 
of the Euclidean space 
is called a root system in 
if:
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 semisimple Lie algebra are 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 symmetry group
of the root system.
