  Discretization methods for ordinary differential equations based on the use
of matrix exponentials have been known for decades. This set of ideas has come
off age and acquired greater urgency recently, within the context of geometric
integration and discretization methods on manifolds based on the use of
Lie-group actions.
  In the present paper we study the approximation of the matrix exponential in
a particular context: given a Lie group $G$ and its Lie algebra $g$, we seek
approximants $F(tB)$ of $\exp(tB)$ such that $F(tB)\in G$ if $B\in g$. Having
fixed a basis of the Lie algebra, we write $F(tB)$ as a composition of
exponentials of the basis elements pre-multiplied by suitable scalar functions.
