A simple algorithm, which exploits the associativity of the BCH formula, and that can be generalized by iteration, extends the remarkable simplification of the BakerCampbell-Hausdorff (BCH) formula, recently derived by Van-Brunt and Visser. We show that if [ X, Y ] = uX + vY + cI , [ Y, Z ] = wY + zZ + dI , and, consistently with the Jacobi identity, [ X, Z ] = mX + nY + pZ + eI , then