 Origin dependence of event-position orthogonality

The Minkowski inner product of two event-position four-vectors is invariant under a common Lorentz transformation fixing the origin, but changes under a common translation. It is therefore not a criterion for the causal separation of the events. For example $(cT,0)$ and $(0,R)$ are orthogonal while their separation has squared interval $c^2{T}^2-R^2$, which can have any sign. Causal classification uses their difference, not their individual positions.