In some calculations, I have arrived at the following algebraic structure, reminiscent of a Lie algebroid, but not quite.

I have a real line bundle $E \to M$, on whose smooth sections $\Gamma(E)$ I have a Lie algebra structure.  I also have a Lie algebra homomorphism $\rho:\Gamma(E) \to \Gamma(TM)$, obeying
$$
[e_1,f e_2] = f [e_1,e_2] + \rho(e_1)(f) e_2
$$
for all sections $e_1,e_2 \in \Gamma(E)$ and functions $f \in C^\infty(M)$.  **However** I do *not* have a Lie algebroid because the map $\rho$ is not $C^\infty(M)$-linear, so it is not induced by a bundle map $E \to TM$.

**Question**: Does such a structure have a name?  Any references where such structure has been studied?

Thanks in advance.