Lie algebra of homogeneous operators of a vector bundle

We prove that for a vector bundle $E\to M$, the Lie algebra ${\mathcal{D}}_{\mathcal{E}}(E)$ generated by all differential operators on $E$ which are eigenvectors of $L_{\mathcal{E}},$ the Lie derivative in the direction of the Euler vector field of $E,$ and the Lie algebra ${\mathcal{D}}_G(E)$ obtained by Grothendieck construction over the $\mathbb{R}-$algebra $\mathcal{A}(E):=\mathrm{P}\mathrm{o}\mathrm{l}(E)$ of fiberwise polynomial functions, coincide up an isomorphism. This allows us to compute all the derivations of the $\mathbb{R}-$algebra $\mathcal{A}(E)$ and to obtain an explicit description of the Lie algebra of zero-weight derivations of $\mathcal{A}(E)\mathrm{.}$