The pair $(E,\varepsilon)$ is determined by $L(E,\varepsilon)$

Let $\mathbb K$ be $\mathbb R$ or $\mathbb C$, let $E\subseteq\mathbb K^N$ be a subspace and let $\varepsilon,\varepsilon'$ be bilinear forms on $E$. Then

1. the projection of $L(E,\varepsilon)$ to $\mathbb K^N$ is exactly $E$;
2. if $L(E,\varepsilon)=L(E,\varepsilon')$ then $\varepsilon=\varepsilon'$.

Together these say that $(E,\varepsilon)\mapsto L(E,\varepsilon)$ is injective, which is the "one-to-one" part of the correspondence between maximal isotropic subspaces and pairs $(E,\varepsilon)$.

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.