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**X**=(cdt, dx, dy, dz), which surely is Lorentz-invariant. I'm talking about the so-called position four-vector,

**X**=(ct, x, y, z). Isn't that just an arrow from an arbitrary origin to a fixed point? Unless all frames decide to use the same origin, which isn't needed for all the other four-vectors, then I don't see how it's Lorentz-invariant.