Lemma: Quadratic-form ambiguity

In coordinates with $K={\mathcal{L}}_m$, a nonzero quadratic form vanishing on the future-null boundary is a scalar multiple of

If it is positive on the cone interior, the scalar is positive.

Proof. Write the form as $a{t}^2+2tb\cdot x+x^{T}Cx$. Evaluation at $(1,u)$ and $(1,-u)$, $\Vert u\Vert =1$, gives $b=0$. Then $u^{T}Cu=-a$ for every unit $u$, so $C=-aI$. The sign follows from evaluation at $(1,0)$.