Suppose $x,y$ are two causal vectors such that $\la x,y\ra=0$. Then $x,y$ are linearly dependent.
If $\dim\lhull{x,y}=2$, then $\lhull{x,y}$ must contain a space-like vector $z=sx+ty$: i.e. $0 < s^2\la x,x\ra+t^2\la y,y\ra\leq 0$.