Almost but not quite a Lie algebroid: what is it?

Question

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.

Edit: I thought it would be instructive to mention that this is a situation which can only arise when $E$ is a line bundle. If $E$ were of higher rank, then $C^\infty(M)$-linearity follows. To see this simply evaluate $[f e_1, g e_2]$ in two different ways: $$ [f e_1, g e_2] = g [f e_1, e_2] + \rho(f e_1)(g) e_2 = f g [e_1,e_2] - g \rho(e_2)(f) e_1 + \rho(f e_1)(g) e_2~, $$ but also $$ [f e_1,g e_2] = -[g e_2, f e_1] = f g [e_1,e_2] + f \rho(e_1)(g) e_2 - \rho(g e_2)(f) e_1~. $$ Equating the two, $$ (\rho(f e_1)-f\rho(e_1))(g)e_2 + (\rho(g e_2)-g \rho(e_2))(f) e_1 = 0~. $$ If $\text{rank} E>1$, then around any point there is a neighbourhood where we can choose $e_1,e_2$ to be linearly independent, whence their coefficients must vanish. Clearly this is not possible for a line bundle.

Answer 1

José, after your last comment, I am pretty sure that you are simply in the presence of a Jacobi structure on a nontrivial line bundle. Most of what I write below is taken from this paper by Crainic and Salazar, called Jacobi structures and Spencer operators.

Just for completeness, given a real line bundle $E \to M$, a Jacobi structure on its space of sections $\Gamma(E)$ is a local Lie bracket $[\cdot,\cdot]$ on $\Gamma(E)$, whereby local means that, for any (possibly locally defined) $e_1,e_2 \in \Gamma(E)$, the support of $[e_1,e_2]$ is contained in the intersection of the supports of $e_1$ and $e_2$. These were originally studied by Kirillov (under the name local Lie algebras) and by Lichnerowicz (first under the assumption that the line bundle be trivial).

Informally speaking, Jacobi structures are to Poisson structures as contact manifolds are symplectic manifolds (this is a very useful analogy if you are to work with these objects). Just like a Poisson structure on a manifold $M$ induces a Lie algebroid structure on $T^*M$, so does a Jacobi structure on $E \to M$ induce a Lie algebroid structure on $J^1 E$, the first jet bundle of $E \to M$. If I am not mistaken, this was first proven by Dazord and is Proposition 3.4 in Crainic and Salazar's paper.

The map $\rho$ that José mentions is simply what Crainic and Salazar call $\rho^1$ in their proof of Proposition 3.4, while the bivector $T_e$ in José's question is $\rho^2(- \otimes e)$ in Crainic and Salazar's language. The relation between the two (which is what José wrote in answer to David's comment) is equation (8) in the paper. The anchor of the Lie algebroid structure on $J^1 E$ is completely determined by $\rho^1$ and $\rho^2$ (this is the content of the first part of the proof of Proposition 3.4 in the paper), so I suspect that José may have just rediscovered this fact!