Reduced and nonreduced presentations of Weyl group elements

This paper is a sequel to work of Dynkin on subroot lattices of root lattices and to work of Carter on presentations of Weyl group elements as products of reflections. The quotients $L/L_1$ are calculated for all irreducible root lattices $L$ and all subroot lattices $L_1$. The reduced (i.e. those with minimal number of reflections) presentations of Weyl group elements as products of arbitrary reflections are classified. Also nonreduced presentations are studied. Quasi Coxeter elements and strict quasi Coxeter elements are defined and classified. An application to extended affine root lattices is given. A side result is that any set of roots which generates the root lattice contains a $Z$-basis of the root lattice.