Cohomology of courant algebroids with split base

In this paper we study the cohomology H^._st(E) of a Courant algebroid E. We prove that if E is transitive, H^. _st(E) coincides with the naive cohomology H^. _naive(E) of E as conjectured by Stiénon and Xu. For general Courant algebroids E we define a spectral sequence converging to H ^._st(E). If E is with split base, we prove that there exists a natural transgression homomorphism T₃ (with image in H ³_naive(E)) which, together with H^. _naive(E), gives all H^._st(E). For generalized exact Courant algebroids, we give an explicit formula for T₃ depending only on the Ševera characteristic clas of E.