Cohomology of Courant algebroids with split base

We study the (standard) cohomology $H_{st}^{\bullet }(E)$ of a Courant algebroid $E$. We prove that if $E$ is transitive, the standard cohomology coincides with the naive cohomology $H_{naive}^{\bullet }(E)$ as conjectured by Stienon and Xu. For a general Courant algebroid we define a spectral sequence converging to its standard cohomology. If $E$ is with split base, we prove that there exists a natural transgression homomorphism $T_3$ (with image in $H_{naive}^3(E)$) which, together with the naive cohomology, gives all $H_{st}^{\bullet }(E)$. For generalized exact Courant algebroids, we give an explicit formula for $T_3$ depending only on the \v{S}evera characteristic clas of $E$.