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 , and with respect to this basis the structure constants of the Lie bracket are given by paths in . This construction is then related to the constructions of Ringel and Peng and Xiao.