Let $(M,g)$ a semi-Riemannian manifold, and let $\mathcal{G}$ a be Lie pseudogroup which acts by local isometries on $(M,g)$. Clearly the germs of all local killing vector fields at $p \in M$ form 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?