On the twisted Dorfman-Courant like brackets

There are completely described allVBm,n-gauge-natural operatorsCwhich, like tothe Dorfman–Courant bracket, send closed linear3-forms $H∈Γl−closE(∧3T∗E)$on a smooth(C∞) vector bundleEintoR-bilinear operatorsCH: $ΓlE(T E⊕T∗E)×ΓlE(T E⊕T∗E)→ΓlE(T E⊕T∗E)$ transforming pairs of linear sections of $T E⊕T∗E→E $ into linear sections of $T E⊕T∗E→E.$ Then all suchCwhich also, like to the twisted Dorfman–Courant bracket, satisfy both some“restricted” condition and the Jacobi identity in Leibniz form are extracted.