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When you derive the Lorentz transformations from a certain set of axioms (possibly Einstein's 1905 axioms, or possibly something less archaic), a typical piece of input to the calculation is that when we do a boost by v, followed by a boost by -v, we have to get an identity transformation: L(-v)L(v)=I.

What is the most fundamental reason why this has to be this way, and how is this derivable strictly from your favorite axiomatization without having to bring in some other ad hoc principle?

You can certainly get some silly results if you don't require this. For example, suppose that I let [itex]L(-v)L(v)=(1+v^2)I[/itex], i.e., a dilation of both time and space by the same factor. This leaves the speed of light (which equals 1) invariant. I could oscillate a meter-stick to the left and right, and as time went on, the meter-stick's length would grow exponentially. This violates time-reversal invariance, but other than that it's not obviously logically impossible. For example, in the Weyl gauge theory, the rate of a clock depends on its past history of motion. It's nutty, but it's not obviously logically impossible.

-Ben