= Courant bracket
{c}
{title2=$[X+\xi,Y+\eta]_C$}
The untwisted skew-symmetric <Courant bracket> on sections of the <generalized tangent bundle> $TM\oplus T^*M$ is
$$
[X+\xi,Y+\eta]_C=[X,Y]+\mathcal L_X\eta-\mathcal L_Y\xi-\tfrac12d\bigl(\eta(X)-\xi(Y)\bigr).
$$
Here $[X,Y]$ is the <Lie bracket of vector fields>, $\mathcal L$ is the <Lie derivative>, and $d$ is the <exterior derivative>. It extends the <Lie bracket of vector fields>, but its Jacobi identity has an exact-form defect; it is not simply another <Lie algebra> bracket.<Lectures on generalized geometry by Nigel Hitchin>[section 1.2, Courant bracket and its Jacobiator]{ref}
For pure <vector fields>, $[X,Y]_C=[X,Y]$, and two pure <one-forms> have zero bracket. On the real line, $[\partial_x,f(x)\,dx]_C=\tfrac12f'(x)\,dx$. This mixed example shows how the bracket couples tangent and cotangent information through differentiation. Omitting the final exact term gives the nonskew Dorfman operation, a related convention with different identities.
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