 A kind of Lie bracket for such formal sums was introduced in 1990 and is called the Courant bracket which is defined by

where formula_15 is the Lie derivative along the vector field "X", "d" is the exterior derivative and "i" is the interior product.A generalized complex structure is a generalized almost complex structure such that the space of smooth sections of "L" is closed under the Courant bracket.There is a one-to-one correspondence between maximal isotropic subbundle of formula_1 and pairs formula_17 where E is a subbundle of T and formula_18 is a 2-form.