In this page there is a reproduction of an argument (I believe originally due to Landau?) that the interval between two events in spacetime is independent of the observer. I'm trying to make more precise mathematical sense of this argument. Here's how I interpret it: the first observer uses his coordinates $(t, x, y, z)$ to write down a signature (1, 3) metric on spacetime $\eta_1 = dt^2 - dx^2 - dy^2 - dz^2$. Similarly, the second observer uses his coordinates $(t', x', y', z')$ to write down the metric $\eta_2 = dt'^2 - dx'^2 - dy'^2 - dz'^2$.
At each point $p$ of spacetime $M^4$, the following condition on a tangent vector $v_p \in T_p M$ should hold: $$ \eta_1(v_p, v_p) = 0 \iff \eta_2(v_p, v_p) = 0 \ \ \ \ \ \ \ \ (*)$$ This just says that observers should agree on which vectors are light-like, i.e. what are the possible velocities of light.
It is then claimed that there must be a proportionality $\eta_1 = a \eta_2$ (a priori possibly depending on p, but in reality not, due to homogeneity assumptions). So this leads me to the question: if $\eta_1, \eta_2$ are two linear signature (1, 3) metrics on a 4-dimensional real vector space which satisfy condition $(*)$, must there be a real constant $a$ such that $\eta_1 = a \eta_2$?
In The Structure of Space-Time Transformations, Borchers & Hegerfeldt show that any self-bijection of Minkowski space (of dimension $\geq 3$) which maps light-cones to light-cones must, up to a scaling factor, be an affine Lorentz transformation.
In my question, letting without loss of generality $\eta_1$ be the standard Minkowski metric in Minkowski 4-space $M^4$, there is an isometry $T : (M^4, \eta_1) \to (M^4, \eta_2)$ since they are metrics with the same signature. The hypothesis $(*)$ of the question guarantees that $T$ maps light-cones to light-cones, and so T must be a scaling of a Lorentz transformation, which means $\eta_2$ must be a scaling of $\eta_1$.