I really like this solution, thanks!

@Deane, I am confused by your most recent comment. 1) Are you implying that if g is a non degenerate matrix, that it must be the matrix corresponding to the metric of flat space time? 2) g is just a matrix, and the entries of g are functions g(i,k) from R4 -> R. I don't think of g as a function on R4, its just a matrix that can be used to represent the inner product as defined in the OP. Has your issue been cleared up by david speyers excellent answer given below? If not, can you please elaborate your objection because I am interested in the problem you see here.

@Michael, the entries of the vectors X are real numbers, and the entries of A should be functions of 4 real parameters. This can be seen by analogy with rotations in three space, for which the lie algebra corresponds to the three principle axis, and the three real parameters which elements of O(3) depend on correspond to rotations about the principle axis.

@michael, yes for sure since that is the case in general relativity.

If you would forgive me for protesting your comment Qiaochu, then I would say that I would not know how to state a "physical principle" to show "that the Boltzmann distribution only depends on the relative energies of the states" without a reference to the energy of particles. The information theoretic approach is useful for quantum mechanics, but,in my opinion, if we want a clear picture in our head of why the Boltzmann distribution is related to the relative energy of states, we must resort to an analogy with the classical mechanics of systems of extremely large numbers of particles.

I am a little concerned about transforming g to the identity matrix with signature (3,1). How can we be sure that the eigenvalue's of g are all 1 or -1?