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I have been studying special relativity from the Gregory Naber's book: "The geometry of Minkowski spacetime" and I found a very strange proof. In Section 2.1, just before of equation 2.1.2. the book says something like this:

But $M$ has a basis of future-directed timelike vectors so it follows that $F$ must satisfy: $$Fx\cdot y=-x\cdot Fy$$ for all $x$ and $y$ in $M$.

Here $M$ represents the four-dimensional Minkowski spacetime. What I find problematic with this is that, according to me, $M$ doesn't have a basis of future-directed timelike vectors. Every basis in M consist in one timelike vector and three spacelike vectors. I will appreciate if somebody explains me if I'm wrong or not and give me a proof in case I'm wrong.

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    $\begingroup$ Hint: Basis vectors don't need to be orthogonal to each other. $\endgroup$ – Chiral Anomaly Aug 25 '19 at 2:56

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