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.

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