Courant Bracket

Question

Let P be a manifold and D be a subbundle of
${TP} $ :=TP⊕T*P ,then how can one prove that Γ(D) ,sections of D, is closed under the Courant bracket? By courant bracket I mean $[[(X,α),(Y,β)]]=([X,Y],(L_Xβ− L_Yα -d(α(Y))$?