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

The flow method for the Baker-Campbell-Hausdorff formula: exact results