  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 gives a
root basis for g. Moreover we show that this root basis gives a purely
combinatorial construction of g, where root vectors correspond to vertices of a
certain quiver $Gammahat$, and show that with respect to this basis the
structure constants of the Lie bracket are given by paths in $Gammahat$. This
construction is then related to the constructions of Ringel and Peng and Xiao.
