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I'm having trouble solving problem 3 at page 105 in Weinberg's book, The Quantum Theory of Fields, Vol. 1:

*Derive the commutation relations for the generators of the Galilean group directly from the group multiplication law (without using our results for the Lorentz group). Include the most general set of central charges that cannot be eliminated by redefinition of the group generators*.

I can do the first two steps of the problem, but I can't figure out how to see whether a central charge can be eliminated or not, and how. I also know what the final result is, i.e. the only central charge left after redefinition of the generators must be

[tex][K_i,P_j]=im\delta_{ij}[/tex]

where m is a parameter that identifies the irreducible representation, the P's are the spatial translations generators, and the K's are the velocity transformations generators.

Any hint?