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 $\mathcal{A}(E):= {\rm Pol}(E)$ 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 $\mathcal{A}(E).$