Z X generated by R. If R is irreducible then there is a unique W -invariant inner product ( , ) on E(R) such that (α,α) = 2 for all short roots α ∈ R. In general, R is a disjoint union of ﬁnitely many irreducible components, each of which has a unique W -invariant inner product as above. Taking the orthogonal sum of these inner products, we obtain a W -invariant inner product ( , ) on E(R) such that the short roots α in any irreducible component satisfy (α,α) = 2. Then a α,β := β,α∨ = 2(β,α) (α,α) for all α,β ∈ R. The integral matrix (a α,β ) α,β∈Π is the symmetrizable Cartan matrix associated to the root datum.