Correct Lorentz part, incorrect affine part

For $\delta \ne 0$, the affine map

has the same Lorentz part as $J$, but it does not fix the particle worldline pointwise. Indeed,

Its time translation cannot be removed by conjugating it to a spinless rotation with a fixed timelike axis, since it has no fixed point.

This algebraic example explains why the certificate checks the whole affine word rather than only its Lorentz part. Time-shifted identifications also appear in the classical discussion of spinning sources DJH1984.