Assume that $\nabla$ is an arbitrary connection on a manifold. Under what conditions, other than LC connections, the binary operator $\nabla_XY -\nabla_YX$ defines a Lie braket on $\chi^{\infty}(M)$?
If it is not true for a general connection, is there a name for this property of connection?
What is an example or non example of a connection $\nabla $, with non trivial torsion, for which $\nabla_XY -\nabla_YX$ satisfy or dissatisfy the Jacobi identity?