  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.
