= Failure of transitivity for mixed-frame simultaneity
In units $c=1$, consider <spacetime events> $O=(0,0)$, $C=(1,2)$ and $D=(1,0)$, with coordinates ordered $(t,x)$. A <Lorentz boost> of speed $1/2$ assigns $t'_O=t'_C=0$, while the original <inertial frame> assigns $t_C=t_D=1$. Nevertheless $O,D$ are <timelike separated>, so no <inertial frame> regards them as simultaneous. Hence the relation “simultaneous in at least one <inertial frame>” is not transitive. This does not contradict transitivity of <Minkowski simultaneity relative to an inertial observer> for one fixed observer.
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