Natural lifts of Dorfman brackets

In this note we prove that, for a vector bundle $E$ over a manifold $M$, a Dorfman bracket on $TM\oplus E^*$ anchored by $\operatorname{pr}_{TM}$ and with $E$ a vector bundle over $M$, is equivalent to a lift from $\Gamma(TM\oplus E^*)$ to linear sections of $TE\oplus T^*E\to E$, that intertwines the given Dorfman bracket with the Courant-Dorfman bracket on sections of $TE\oplus T^*E$. This shows a universality of the Courant-Dorfman bracket, and allows us to caracterise twistings and symmetries of transitive Dorfman brackets via the corresponding lifts.