Lemma: Independent component checks

The contraction gives

All other jet components are zero in each check.

Proof. The contraction tables give

Writing $t=\mathrm{tr}(\eta H)$, they also give

For $J_{11}=1$, equation (eq:Qmatrix) yields

Its contraction is

so ${\mathcal{K}}^{00}=14$.

For $J_{01}=J_{10}=1$,

The three groups contract to $9,-2,-14$, respectively. Thus the result is $2(9-2-14)=-14$.

For $J_{00}=1$,

Its contraction is

so ${\mathcal{K}}^{22}=10$.