Lie brackets of automorphisms

Let $F$ be the vector fields of a differential manifold $M$, let $[X,Y]$ be the Lie brackets of $F$, now let $a$ be an automorphism of $F$ for the structure of real vector space of $F$. I consider now the bracket: $$[X,Y]_a= a^{-1}[a(X),a(Y)]$$ A simple calculus shows that it satisfies the Jacobi identities, so $(F,[,]_a)$ is a new Lie algebra. My question is: when does this new Lie bracket come from the vector fields of a manifold $M_a$? For example, if I take $a=g_*$, with $g$ a diffeomorphism of $M$, I could say that $M_{g_*}=M$. Perhaps that a condition of a certain smoothness could be added for the automorphism $a$?

By Purcell and Shanks [Shanks, M.E., Pursell, L.E.: The Lie algebra of a smooth manifold. Proc. Amer. Soc. 5, 468-472 (1954)] the algebraic structure of the Lie algebra of smooth vector fields on a smooth manifold uniquely determines the manifold. Namely, a maximal ideal is exactly the set of vector fields which vanish to infinite order at some point, in the $C^\infty$ and compact case. So for $(F,[\;,\;]_a)$ you can construct a manifold structure on the space of maximal ideals, and then $a = f^*$ for a diffeomorphism between the original manifold and the new one.

See also the remarkable paper [J. Grabowski: Isomorphisms and Ideals of the Lie Algebras of Vector Fields, Inventiones math. 50, 13-33 (1978)] which extends this to the Lie algebras of real analytic and holomorphic vector fields and the non-compact case.

• Perhaps, my automorphism $a$ is not smooth; for example I can take a vector field $X$ and define: $a(X)=-X$ and $a(Y)=Y$ otherwise with a choice of a basis of $F$. It doesn't seem to come from any diffeomorphism or manifold. – Antoine Balan Jul 9 '18 at 14:12
• The two manifolds $M$ and $M_a$ are perhaps not diffeomorphic if $a$ doesn't exchange the maximal ideals? $a$ is not an isomorphism of Lie algebras. – Antoine Balan Jul 9 '18 at 14:49
• I apologize, $a$ is an isomorphism of Lie algebras between $(F,[,])$ and $(F,[,]_a)$. Also, all is well defined. – Antoine Balan Jul 9 '18 at 14:57
• Another question: is it also the case if we take the Poisson brackets $\{ f,g \}= \omega (df,dg)$ of functions over a symplectic manifold? The maximal ideals are the points too? – Antoine Balan Jul 9 '18 at 20:26