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.
