Minkowski simultaneity relative to an inertial observer ($\eta (u,Q-P)=0$)

In Minkowski spacetime of metric signature $(-,+,+,+)$, let $u$ be a constant future-directed four-velocity with $\eta (u,u)=-c^2$. The time difference assigned to two spacetime events is $\Delta t_u=-\eta (u,Q-P)/c^2$. Thus the observer's simultaneous events form an affine hyperplane orthogonal to $u$. This is an invariant relation of the three arguments $u,P,Q$; the Minkowski metric alone does not single out one $u$. For fixed $u$, it defines an equivalence relation and the hyperplanes form a foliation. Changing $u$ changes this foliation.

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 $t_O^{\prime }=t_C^{\prime }=0$, 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.