Let M be a manifold.

To what extent all Lie algebra structures with tensorial property on $\chi^{\infty}(M)$ are studied?

That is a Lie algebra structure for which $[X,fY]=f[X,Y]$.

(For every parallelizable $n$-manifold $M$ and every Lie algebra structure on $\mathbb{R}^{n}$, one can introduce a Lie algebra structure on $\chi^{\infty}(M)$ with this tensorial property. So what is an example of non parallelizable case?)

Any such structure on a manifold is a vector valued 2-form, that is an element of $\Omega_{1,2}(M)$. Such vector valued 2-forms are called Jacobi 2-form. The Jacoby identity on $\Omega_{1,2}(M)$ is denoted by $I_{2}$

What identities $I_{1}$ and $I_{3}$ can be defined on $\Omega_{1,1}(M)$ and $\Omega_{1,3}(M)$ such that they are invariant under the differentiation? That is: The differential of every $I_{2}$- form, satisfies $I_{3}$ or the differential of every $I_{1}$-form satisfy $I_{2}$(The Jacoby I dentity). Can one extend these (possible) identities for arbitrary $k$ on $\Omega_{1,k}$?(Then study the subcomplex and its cohomology of vector valued differential forms which satisfy $I_{k}$, for all $k$? May be Such cohomology can be named "Jacobi cohomology")

For compact manifolds, may be some other questions which can be asked in this context are following:

1) Are there some obstructions for existence of a Jacobi 2-form which fibre wise structure is a simple Lie algebra?

2)Is there an example of a Jacobi 2-form such that we obtain different(perhaps infinit number) Lie algebra structures in tangent spaces?