In the lecture notes A brief introduction to Dirac manifolds, 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 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.

In the lecture notes A brief introduction to Dirac manifolds, 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 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.