We have recently shown that the factorizaton of certain Lie algebraic evolution operators into a convergent infinite product of simple evolution operators is possible for one-dimensional cases. In this paper, we deal with the multivariable case. To this end, we formulate the factorization for the general case, then we show that most of the practical problems can be brought to a canonical one. The canonical problem has nothing different in concept but the relevant partial differential equations need not be solved. Three simple illustrative examples and concluding remarks complete the work.