Lemma: Admissibility

All three labels belong to $\mathcal{X}$, with

Proof. The momenta are nonzero, null, and sum to zero. Their pair sums give the stated invariants. Both tensors are real, symmetric, temporally zero, and Lorentz trace free. The spatial support of $k_2$ is in direction three, where $A$ has a zero row. The spatial supports of $k_3,k_4$ are in direction one, where $B$ has a zero row. Thus the momentum and reference transversality conditions hold. Finally ${t_0}^2=-1$, and both gauge vectors are admitted.