  In this article we continue the study of couplings of subelliptic Brownian
motions on the subRiemannian manifolds SU (2) and SL(2, R). Similar to the case
of the Heisenberg group, this subelliptic Brownian motion can be considered as
a Brownian motion on the sphere (resp. the hyperbolic plane) together with its
swept area modulo 4$\pi$. Using this structure, we construct an explicit non
co-adapted successful coupling on SU (2) and, under strong conditions on the
starting points, on SL(2, R) too. This strategy uses couplings of Brownian
bridges, taking inspiration into the work from Banerjee, Gordina and Mariano
[3] on the Heisenberg group. We prove that the coupling rate associated to
these constructions is exponentially decreasing in time and proportionally to
the subRiemannian distance between the starting points. We also give some
gradient inequalities that can be deduced from the estimation of this coupling
rate.
