  The Lie groups $SU(2)$ and $SL(2,\mathbb{R})$ can be viewed as model spaces
in subRiemannian geometry. Coupling two subelliptic Brownian motions on $SU(2)$
(resp. $SL(2,\mathbb{R})$) consists in coupling two Brownian motions on the
sphere (resp. the hyperbolic plane) and simultaneously their swept areas. Using
this approach we propose an explicit construction of a co-adapted successful
coupling on $SU(2)$. The strategy is to alternate between reflection and
synchronous (with noise) coupling on the sphere. We also describe some more
general constructions of co-adapted couplings on $SU(2)$ and also on
$SL(2,\mathbb{R})$.
