  We investigate Birkhoff (or triangular) factorization and (what we propose to
call) root subgroup factorization for elements of a noncompact simple Lie group
$G_0$ of inner type. For compact groups root subgroup factorization is related
to Bott-Samelson desingularization, and many striking applications have been
discovered by Lu (\cite{Lu}). In this paper, in the inner noncompact case, we
obtain parallel characterizations of the Birkhoff components of $G_0$ and an
analogous construction of root subgroup coordinates for the Birkhoff
components. As in the compact case, we show that the restriction of Haar
measure to the top Birkhoff component is a product measure in root subgroup
coordinates.
