  Let U/K represent a connected, compact symmetric space, where theta is an
involution of U that fixes K, phi: U/K to U is the geodesic Cartan embedding,
and G is the complexification of U. We investigate the intersection of phi(U/K)
with the Bruhat decomposition of G corresponding to a theta-stable triangular,
or LDU, factorization of the Lie algebra of G. When g in phi(U/K) is generic,
the corresponding factorization g=ld(g)u is unique, where l in N^-, d(g) in H,
and u in N^+. In this paper we present an explicit formula for d in Cayley
coordinates, compute it in several types of symmetric spaces, and use it to
identify representatives of the connected components of the generic part of
phi(U/K). This formula calculates a moment map for a torus action on the
highest dimensional symplectic leaves of the Evens-Lu Poisson structure on U/K.
