  In this paper, we efficiently compute overapproximated reachable sets for
control systems evolving on Lie groups, building off results from monotone
systems theory and geometric integration theory. We propose to consider
intervals living in the Lie algebra, which through the exponential map,
describe real sets on the Lie group. A local equivalence between the original
system and a system evolving on the Lie algebra allows existing interval
reachability techniques to apply in the tangent space. Using interval bounds of
the Baker-Campbell-Hausdorff formula, a Runge-Kutta-Munthe-Kaas reachability
algorithm is proposed, providing reachable set estimates for arbitrary time
horizons at little computational cost. The algorithm is demonstrated on through
consensus on a torus and attitude control on $SO(3)$.
