Let U∕K represent a connected,
compact symmetric space, where 𝜃 is an involution of U that fixes K, ϕ : U∕K → U
is the geodesic Cartan embedding, and G is the complexification of U. We investigate
the intersection of ϕ(U∕K) with the Bruhat decomposition of G corresponding to a
𝜃-stable triangular, or LDU, factorization of the Lie algebra of G. When
g ∈ ϕ(U∕K) is generic, the corresponding factorization g = ld(g)u is unique, where
l ∈ N− , d(g) ∈ H, and u ∈ N+. 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
ϕ(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.
