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 $a$?
 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.
Added:
For Poisson brackets see the paper 
[MR0438390 (55 #11304a) 
Kirillov, A. A.
Local Lie algebras. (Russian) 
Uspehi Mat. Nauk 31 (1976), no. 4(190), 57–76. 
58F05 (17B65) ] and papers citing this paper. 
