B-field shifts are isometries

Let $B$ be a real 2-form on $T=\mathbb R^N$ (a skew-symmetric matrix, $B^{\mathsf T}=-B$) and let $e^B(X+\xi)=X+\xi+i_XB$. Then for all $u,v\in T\oplus T^*$,

So B-field shifts preserve the natural inner product of the generalized tangent space.

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.