In the lecture notes [A brief introduction to Dirac manifolds][1], Henrique Bursztyn
 recall the notion of a Courant bracket on section of the generalised tangent bundle $TM\oplus T^*M$.

For $X+\xi, Y+\eta\in \Gamma(M,TM\oplus T^*M)$, the Courant bracket is given by 

$$[X+\xi, Y+\eta]=[X,Y]+L_X\eta-L_Y\xi+\frac{1}{2}d(\xi(Y)-\eta(X)).$$

It also mentions that, 

> One may alternatively use, instead of (3.2), the non-skew-symmetric bracket
$$((X,\alpha),(Y,\beta))\mapsto ([X,Y],L_X\beta − i_Y d\alpha)$$ for condition (ii); (3.2) is the skew-symmetrization of this bracket, and a simple computation shows that both brackets agree on sections of subbundles satisfying (i).


In the lecture  [Introduction to Poisson geometry and Lie algebroids][2] Eckhard Meinrenken defines (7 mins to 10 mins) Courant bracket as $((X,\alpha),(Y,\beta))\mapsto ([X,Y],L_X\beta − i_Y d\alpha)$ instead of $[X+\xi, Y+\eta]\mapsto ([X,Y], L_X\eta-L_Y\xi+\frac{1}{2}d(\xi(Y)-\eta(X))$.

He said the following:

> "a weird formula which takes some time to get used to". 

> "I prefer to work with the bracket that is not skew-symmetric."

> "There are some ways of motivating why this is the right  formula, but we do not have time to do that"


So, I would like to understand the motivation in considering this non skew-symmetric bracket.

Please suggest some  references that gives motivation for this.

  [1]: https://arxiv.org/pdf/1112.5037.pdf
  [2]: https://mediaserver.unige.ch/play/95558