Coxeter elements and root bases

Let g be a Lie algebra of type A, D, E with fixed Cartan subalgebra h, root system R and Weyl group W. We show that a choice of Coxeter element C∈W gives a root basis for g. Moreover, using the results of Kirillov and Thind (2010) [KT] we show that this root basis gives a purely combinatorial construction of g, where root vectors correspond to vertices of a certain quiver Γ̂, and with respect to this basis the structure constants of the Lie bracket are given by paths in Γ̂. This construction is then related to the constructions of Ringel and Peng and Xiao.