Minkowski space/Observer vector/Decomposition/Exercise

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

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

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