I am reading the lecture notes on generalized geometry
http://www.staff.science.uu.nl/~caval101/homepage/Research_files/australia.pdf
but now I am stuck on the exercise 1.31 at page 13.
Here is the explanation of the exercise: Given a Courant algebroid $E$ over a manifold M, and given a submanifold $i : N \to M$, we are interested in studying the pullback of $E$ to $N$. The exercise requires to show that, provided two sections of $E$ , called $e_1$ and $e_2$, such that $$\pi (i^\ast e_1) \in \Gamma(TN)$$ ($\pi$ is the anchor) and $$i^\ast e_2 = 0$$, then $$i^\ast [e_1 , e_2]$$ ($[\cdot , \cdot]$ is the Courant bracket on $E$) is not necessarily zero, but lies in the conormal bundle of $N$.
Unfortunately I am not able to show this statement, even in the simplest case in which the algebroid $E$ is just the tangent bundle $TM$, the anchor is just the identity, and the Courant bracket is just the ordinary Lie bracket on vector fields. In this simplified case I am finding that the Lie Bracket would be simply zero.
Thank you very much to everyone will help me to understand the proof of this statement.