Question: commutator of differential operators

I am trying to compute commutator like this [d_t +J_x,d_x+J_t], where d_t and d_x are differential operators, J_x and J_t, which contains generators of group, are some currents on group manifold. terms like this [J_x,J_t] could sovled by physics package, but how to manipulate [d_t,J_x]? How to define operator d_t in order to make the commutator works? and if it works we might have another problem as [d_t,J_x]=[d_t, J^a_x*T_a], where T_a are generators, the result is T_a *d_t(J^a_x)-J^a_x*T_a*d_t, and the second term will disapear when we consider the commutator as operator, like in quantum mechanics. So how could we let the computer now this? 

possible useful packages are PDEtools,Ore_algebra, physics,

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