  This paper is a sequel to [Caine A., Pickrell D., arXiv:0710.4484], where we
studied the Hamiltonian systems which arise from the Evens-Lu construction of
homogeneous Poisson structures on both compact and noncompact type symmetric
spaces. In this paper we consider loop space analogues. Many of the results
extend in a relatively routine way to the loop space setting, but new issues
emerge. The main point of this paper is to spell out the meaning of the
results, especially in the SU(2) case. Applications include integral formulas
and factorizations for Toeplitz determinants.
