Properties of Simple Roots

Consider a given abstract root system Δ and some simple root system Π ⊂ Δ. We may represent β ∈ Δ as a linear combination of elements of Π with integer coefficients of the same sign. By definition, we know that the elements of Π are linearly independent. Thus, such a representation of β is unique and exactly one of ±β is a positive root associated to Π. Δ+ is simply the set of all positive roots associated to Π, and so our choice of simple root system determines an associated set of positive roots uniquely. Now suppose instead that we are given a positive root system Δ+ ⊂ Δ. We can construct a vector t ∈ V such that �t, α� = 0 for all α ∈ Δ, and such that Δ+ = Δ t , where Δ t = {α ∈ Δ : �t, α� > 0}. We can be sure from [3] that such a vector always exists, and by Definition 4 in [3], we can determine the set of all indecomposable elements in Δ . Lets t call this set Πt. By Theorem 9 in [3], Π = Πt, and we have now proven our first theorem.