Motivated by this MO question we ask the following two questions:

1)What is an example of a compact manifold $M$ which does not admit any smooth (1,2) tensor $\omega$ which restriction to each fibre(tangent space) gives a simple Lie algebra?

2)What is an example of a compact manifold $M$ which admit at least one smooth (1,2) tensor $\omega$ which restriction to each fibre(tangent space) gives a simple Lie algebra?