Let $(M,g)$ a semi Riemannian manifold and let $\mathcal{G}$ a Lie pseudogroup that acts by local isometries on $(M,g)$. Clearly the germs of all local killing vector fields at $p \in M$ forms a lie algebra. What if I restrict myself to germs of those local killing fields, whose local flow is contained in $\mathcal{G}$? Do they form a lie algebra as well? If so, why?