Type-$0$ structures are B-shifts of symplectic ones

Let $\mathcal J$ be a generalized almost complex structure on $T\oplus T^*$ whose $\sqrt{-1}$-eigenbundle has type $0$. Then there are a real 2-form $B$ and a nondegenerate real 2-form $\omega$ on $T$ such that

This is the pointwise content of the statement that, up to a shift of the B-field, symplectic structures are the only type-$0$ generalized complex structures.

Formalization Note Pointwise, no closedness of $B$ or $\omega$ is involved; the global statement in the source additionally needs integrability.

Throughout, $T=\mathbb R^N$ models a tangent space of an $N$-manifold, $T^*$ its dual (identified with $\mathbb R^N$ via the standard dual basis), and $T\oplus T^*$ the generalized tangent space with the natural inner product $\langle X+\xi,Y+\eta\rangle=\tfrac12(\xi(Y)+\eta(X))$. A generalized almost complex structure is a real linear map $\mathcal J$ of $T\oplus T^*$ with $\mathcal J^2=-\mathrm{Id}$ and $\langle\mathcal J u,\mathcal J v\rangle=\langle u,v\rangle$; its $\sqrt{-1}$-eigenbundle is $L=\{v\in(T\oplus T^*)\otimes\mathbb C:\mathcal Jv=\sqrt{-1}\,v\}$.

Formalization Note The statement is fiberwise: $T\oplus T^*$ is the coordinate space GIdx N → ℝ (vector part on Sum.inl, one-form part on Sum.inr), endomorphisms are matrices indexed by GIdx N, and the complexification is GIdx N → ℂ with the real matrix acting through the coercion $\mathbb R\to\mathbb C$. All notions are those of the shared definition file Definitions.Def_GeneralizedComplex_Basic.