The pure-gauge two-jet

Testing only $h=K\xi$ with $k=0$ is not a second-order pure-gauge jet: the straight-line family $\eta +\kappa K\xi$ is not pure gauge through second order, and ${\mathsf{X}}_2[K\xi ,0]=\frac{1}{2}\xi \cdot \partial \xi -{\mathsf{X}}_1[\frac{1}{2}{\mathcal{L}}_{\xi }K\xi ]$ by (eq:X2affine).