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There's nothing privileged.
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It's the one with the minus signs in it
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that has the privileged role
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of playing the role of the dot product.
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So, space-time is not Euclidean.
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Euclidean space is the space in which we live,
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and distance squared is the sum of the squares of all the coordinates.
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So, it's a pseudo-Euclidean space
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in which to find the invariant length,
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you've got to square some components
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and subtract from them the square of some other components.
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Okay. So, finally,