  In previous work we proved that for a SU(2,C) valued loop having the critical
degree of smoothness (one half of a derivative in the L^2 Sobolev sense), the
following are equivalent: (1) the Toeplitz and shifted Toeplitz operators
associated to the loop are invertible, (2) the loop has a triangular
factorization, and (3) the loop has a root subgroup factorization. For a loop g
satisfying these conditions, the Toeplitz determinant det(A(g)A(g^{-1})) and
shifted Toeplitz determinant det(A_1(g)A_1(g^{-1})) factor as products in root
subgroup coordinates. In this paper we observe that, at least in broad outline,
there is a relatively simple generalization to loops having values in SL(2,C).
The main novel features are that (1) root subgroup coordinates are now rational
functions, i.e. there is an exceptional set, and (2) the non-compactness of
SL(2,C) entails that loops are no longer automatically bounded, and this
(together with the exceptional set) complicates the analysis at the critical
exponent.
