Proposition: Boosts select the cone

A proper cone satisfying $\mathsf{B}$, with $\dim W\ge 1$, equals $\{t{e}_0+x:t\ge \Vert x{\Vert }_2\}$.

Proof. Positive multiples of $B_n(s)e_0$ fill the future timelike region. Closedness therefore gives $L_m\subseteq K$.

Suppose $v=(t,x)\in K\setminus L_m$. If $x=0$, then $t<0$, contradicting pointedness because $e_0\in K$. Otherwise set $a=\Vert x{\Vert }_2>0$, $n=x/a$. Then $t<a$, and invariance and scaling give $e^s{B}_n(s)v\in K$. As $s\to -\infty$,

This is a negative nonzero multiple of a vector already in $L_m\subseteq K$, again contradicting pointedness.