Many applications in science and engineering involve tracking the state of a stochastic differential equation (SDE) evolving in a Lie group. This has been tackled by particle filtering although some existing schemes fail to satisfy geometric constraints. Further the conventional particle filter suffers from particle deprivation. Here we overcome these problems by managing the geometry with the Cayley transform and particle depletion with optimal transport. Simulations show the superiority over 'regular' geometry aware particle filtering.