Lemma: Spinless matching

For $0<\mu <2\pi$, the fixed-point set of $J(\mu )$ is ${\mathcal{W}}_0$, its Lorentz part fixes $\mathbf{t}$, and its oriented rest-frame spatial rotation is $\mu$. Consequently $H_A(\mu )$ fixes $A{\mathcal{W}}_0$ pointwise and its Lorentz part fixes $C\mathbf{t}$.

For $\delta \ne 0$, the map $(R(\mu ),\delta \mathbf{t})$ has no fixed point and is not an admitted spinless peripheral map.

Proof. The spatial rotation has no nonzero fixed spatial vector for $0<\mu <2\pi$, while its time component is unchanged. Affine conjugation transports the fixed set. A fixed point of $(R(\mu ),\delta \mathbf{t})$ would require $T=T+\delta$, which is impossible.