In this question we try to improve some parts of [this post](https://mathoverflow.net/questions/209694/differential-and-pre-differential-of-jacobi-identity) as follows:



> 
> What is an example of a manifold $M$ and a Lie algebra $L$ (with the same dimension) such that $M$ **does not** admit an smooth $(1,2)$ tensor $\alpha$ which satisfies the Jacobi identity at each fibre $T\_x(M)$ and the fibrewise Lie algebra structure $\alpha\_x$ on $T\_x (M)$ is isomorphic to $L$?
> 
> 
> 


What type of obstructions would appear?



Manifold_Lie algebra compatibility
A simple example is to let $M=S^2$ and let $L$ be the nonabelian Lie algebra of dimension $2$. If such an $\alpha$ existed, its range would be a rank-1 subbundle $L\subset TS^2$, but this cannot exist for topological reasons.


The general obstruction is whether the manifold $M$ admits an $\mathrm{Aut}(L)$-structure, and this can be determined by homotopy-theoretical methods. 


Another good example is to take $M=S^4$ and $L$ any nonabelian Lie algebra of dimension~$4$. Since $\mathrm{Aut}(L)$ always preserves a nontrivial subalgebra of $L$ in this case, such a structure $\alpha$ on $M$ would induce a nontrivial subbundle of $TS^4$, and this does not exist, so $\alpha$ cannot exist. 


