Minkowski space/Observer vector/Decomposition/Exercise

Let ${}V$ be a Minkowski space with the Minkowski form ${}\left\langle -,-\right\rangle$. Show that for every observer vector ${}v\in V$, there exists a direct sum decomposition

${}V=\mathbb {R} v\oplus (\mathbb {R} v)^{\perp }\,,$

where the restriction of the Minkowski form to ${}\mathbb {R} v$ is negative definite, and the restriction of the Minkowski form to ${}(\mathbb {R} v)^{\perp }$ is positive definite.

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