  We develop non-Markovian, non-co-adapted couplings for sub-Riemannian
Brownian motions in various sub-Riemannian manifolds, namely the
three-dimensional Heisenberg group, higher-dimensional non-isotropic Heisenberg
groups, SL(2,R) and its universal cover, and SU(2,C). Our primary focus is on
the situation when the processes start from two points on the same vertical
fiber, since in general co-adapted couplings cannot give the sharp rate for the
coupling time in this case. Then for general points, we use this vertical
coupling as the second stage of a two-stage coupling. Non-Markovian couplings
in this context were first introduced by Banerjee-Gordina-Mariano, for the
three-dimensional Heisenberg group, and were more recently extended by
B\'en\'efice to SL(2,R) and SU(2,C), using a detailed consideration of the
Brownian bridge. In contrast, our couplings are based on global isometries of
the space, giving couplings that are maximal, as well as making the
construction relatively simple and uniform across different manifolds.
Moreover, this construction gives a coupling time that equals a hitting time
for the vertical component of one of the Brownian motions, and this vertical
component satisfies a reflection principle, which is useful in explicitly
bounding the tail probability of the coupling time. We estimate the coupling
time in these various situations and give applications to inequalities for the
heat semigroup.
