  Mostow's Decomposition Theorem is a refinement of the polar decomposition. It
states the following. Let G be a compact connected semi-simple Lie group with
Lie algebra g. Given a subspace h of g such that [X, [X, Y]] belongs to h for
all X and Y in h, the complexified group G^C with Lie algebra g + ig is
homeomorphic to the product G .exp im. exp ih, where m is the orthogonal of h
in g with respect to the Killing form. This Theorem is related to geometric
properties of the non-positively curved space of positive-definite symmetric
matrices and to a characterization of its geodesic subspaces. The original
proof of this Theorem given by Mostow uses the compactness of G. We give a
proof of this Theorem using the completeness of the Lie algebra g instead,
which can therefore be applied to an L*-group of arbitrary dimension. Some
applications of this Theorem to the geometry of stable manifolds and affine
coadjoint orbits are given.
