# Do the germs of local vector fields which are generated by a Lie pseudogroup form a Lie algebra?

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?

• Have you tried to use the definition of a Lie bracket using (local) flows? It seems the answer is yes because all steps needed to set up the Lie bracket this way can be performed in $\mathcal G$. – Sebastian Goette Jan 28 '16 at 15:17
• Hi @Sebastian. I know how to calculate the commutator pointwise in terms of local flows. But I need the local flow of the commutator vector field to be contained in $\mathcal{G}$. I do not see why this is true, although I guess it is. – hase_olaf Jan 29 '16 at 10:34
• Let $\Phi_t$, $\Psi_t$ be the local flows. I think the curve $\Phi_{\sqrt t}\Psi_{\sqrt t}\Phi_{-\sqrt t}\Psi_{-\sqrt t}$ in $\mathcal G$ is differentiable from above in $t$ at $t=0$ and has the Lie bracket as differential. – Sebastian Goette Jan 29 '16 at 13:30
• @Sebastian: How can a curve in $\mathcal G$ be differentiable? As $\mathcal G$ is merely a Lie pseudogroup, it does not possess a differentiable structure. Thanks for helping! – hase_olaf Jan 30 '16 at 10:50
• I don't know what the "Lie" in "Lie pseudogroup" stands for. Nevertheless, the local flow of the Lie bracket will be contained in the closure of $\mathcal G$ in an appropriate sense, by arguments similar to my comments. So if you demand or prove that $\mathcal G$ is closed, you can get those commutators. – Sebastian Goette Feb 1 '16 at 12:58