Suppose $\dim F\geq2$, then $F$ is time-like iff $F$ contains two linearly independent light-like vectors.
Let $x,y\in F$ be two linearly independent light-like vectors, then $\la x,y\ra\neq0$ by exam and thus we may assume that $\la x,y\ra < 0$. Hence
$$
\la x+y,x+y\ra=2\la x,y\ra < 0~.
$$