Minkowski-space/Observer vectors/Fact

Let ${}V$ be a Minkowski space, endowed with a Minkowski form ${}\left\langle -,-\right\rangle$. Then the following statements hold.

1. For every observer vector ${}v\in V$, we have 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 where the restriction of the Minkowski-form to ${}V_{v}=(\mathbb {R} v)^{\perp }$ is positive definite. Here, ${}V_{v}$ consists of spacelike vectors.

2. For two observer vectors ${}v,w\in V$ from the same half cone, we have

   ${}\left\langle v,w\right\rangle <0\,.$

3. For timelike vectors ${}v,w\in V$, we have

   ${}\left\langle v,w\right\rangle ^{2}\geq \left\langle v,v\right\rangle \cdot \left\langle w,w\right\rangle \,.$

Write a proof