Let $M$ be a smooth manifold and $\tau M$ its second-order tangent bundle. A second-order vector field $A\in \Gamma(\tau M)$ can locally be expressed as a finite sum of operators $C^\infty(M)\rightarrow C^\infty(M)$ of the form
where $L_V:C^\infty(M)\rightarrow C^\infty(M)$ denotes the Lie derivative along the vector field $V$.
The smooth functions can be regarded as a subspace of the exterior algebra over $M$. Moreover, the Lie derivative defined on this subspace can be extended to the entire exterior algebra by requiring $L_V$ to be a wedge-product derivation that commutes with the exterior derivative. Thus, an operator $C^\infty(M)\rightarrow C^\infty(M)$ of the form $L_X+L_YL_Z$ extends to the entire exterior algebra as well.
Do second-order vector fields determine differential operators on the exterior algebra in a natural way? Given a decomposition of $A$ as a sum of operators of the form given above, the answer is yes. I'm having trouble understanding if the particular decomposition really matters.