Leveraging techniques from the literature on geometric numerical integration,
we propose a new general method to compute exact expressions for the BCH
formula. In its utmost generality, the method consists in embedding the Lie
algebra of interest into a subalgebra of the algebra of vector fields on some
manifold by means of an isomorphism, so that the BCH formula for two elements
of the original algebra can be recovered from the composition of the flows of
the corresponding vector fields. For this reason we call our method the flow
method. Clearly, this method has great advantage in cases where the flows can
be computed analytically. We illustrate its usefulness on some benchmark
examples where it can be applied directly, and discuss some possible extensions
for cases where an exact expression cannot be obtained.
