A common generalization of the Fr\"olicher-Nijenhuis bracket and the Schouten bracket for symmetry multi vector fields

There is a canonical mapping from the space of sections of the bundle $\wedge T^\ast M\otimes ST\ M$ to $\Omega(T^\ast M ; T(T^\ast M))$. It is shown that this is a homomorphism on $\Omega(M;TM) for the Fr\"olicher-Nijenhuis brackets, and also on $\Gamma(ST\ M)$ for the Schouten bracket of symmetric multi vector fields. But the whole image is not a subalgebra for the Fr\"olicher-Nijenhuis bracket on $\Omega(T^\ast M;T(T^\ast M))$.