  Root subgroup factorization is a refinement of triangular (or LDU)
factorization. For a complex reductive Lie group, and a choice of reduced
factorization of the longest Weyl group element, the forward map from root
subgroup coordinates to triangular coordinates is polynomial. We show that the
inverse is rational. There is an algorithm for the inverse (involving LDU
factorization), and a related explicit formula for Haar measure in root
subgroup coordinates. In classical cases there are preferred reduced
factorizations of the longest Weyl group elements, and conjecturally in these
cases there are closed form expressions for root subgroup coordinates.
