The causal cones are convex

Question

Taking into account the definitions of causal, temporal, spatial and luminous vectors, I want to test the following results:

1. Each temporal cone is convex.

The proof of this first result would be the following: Let v, w be temporal in the same cone, and a, b> 0 then, using that two temporal vectors v and w fall in the same temporal cone if and only if g (v, w) <0. We have the following: $$g(v, av + bw) = ag(v, v) + bg(v, w) <0, g(av + bw, av + bw) <0.$$ From the last inequality, av + bw is temporary, and from the first it falls on the same cone as v.

2. The causal cones are convex.

In this case I am unable to perform the test correctly. I think you would have to use some of the properties like that:

i) If u, v $\in$ V are two linearly independent causal vectors, then u and v are in the same cone if and only if $g (u, v) <0$.

ii) Given two causal vectors u, v $\in$ V, then $g (u, v) \neq 0 $ unless u and v are both luminous and linearly dependent.

Could you help me with proof 2, that is, the causal cones are convex?