I am writing something for the journal of the university on lorentz space, and I want to prove that the following definition of the hyperbolic distance on the upper sheet of the hyperboloid $v \cdot v=-1$ is a metric:

The hyperbolic distance between $u$ and $v$ is the only positive number $\eta (u,v)$ such that $cosh(\eta(u,v))=-u \cdot v$.

The first properties are easy, but all the proofs of the triangle inequality that I have seen seem complicated. In Ratcliffe's Foundations of Hyperbolic Manifolds the lorentzian cross product is used along with many of its properties, and I would like not having to introduce the cross product just for this proof.

I have thought about showing that the hyperbolic angle is the same as the lorentzian arc-length distance, and then showing that the metric given by the arc length satisfies the triangle inequality, but this seems even more tedious. Is there a shorter and nicer proof of this?