Let us consider a manifold M, subject to the action of a Lie group G. The simplest instance of such situation is when M is a Lie group itself. Given a vector field on M, that can be described suitably via the prescribed Lie group action, it is possible to construct on M a numerical approximation of its corresponding flow, via a Lie group method. In order to do that a crucial tool to be provided is a coordinate map (local diffeomorphism) from the Lie algebra of the Lie group G to G. The exponential map is one possible choice, but its numerical approximation needs to be "exact" to machine precision in order to preserve in the numerical solution the geometric features of the exact solution of the ode in question.
Especially in the case of matricial Lie groups, computationally cheaper coordinate maps are available. We will discuss some strategies based on the use of splitting techniques and canonical coordinates of the second kind for the approximation of the matrix exponential. Comparisons of the methods with implementations based on the use of traditional methods of approximation of the matrix exponential will be shown in some numerical experiments that conclude the talk.