The classification of root systems

2 Root systems Definition. Let E ∼= R be a vector space with an inner product 〈·, ·〉. A subset R ⊂ E \ {0} is called root system, if R has the following properties: (R1) R is finite and spans E, (R2) if α ∈ R, then −α ∈ R and ±α are the only multiples of α in R, (R3) R is invariant under the reflection in the hyperplane orthogonal to any α ∈ R (see Fig. 1), i.e., for all α, β ∈ R: sα(β) = β − 2 projα β ∈ R, (1) where projα β is the projection of β on α: projα β = α 〈β, α〉 〈α, α〉 , (2)