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Does somebody have some interesting examples of Courant algebroids which are not exact? By exact I mean one which is of the form $TM\oplus T^\star M$ with the standard bracket twisted by a closed 3-form $H$.

Thank you!

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The answer depends on what you mean by ``interesting."

For example the paper "On the Geometric Structure of Hamiltonian Systems with Ports" by Jochen Merker, J Nonlinear Sci (2009) 19: 717–738 DOI 10.1007/s00332-009-9052-3, may be considered as dealing with interesting examples of Courant algebroids. The algebroids there are not of the form $TM\oplus T^*M \to M$.

Grützmann's thesis (arXiv:1004.1487 [math.DG]) maybe another good place to look.

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