$L(E,\varepsilon)$ is maximal isotropic

Let $\mathbb K$ be $\mathbb R$ or $\mathbb C$, let $E\subseteq\mathbb K^N$ be a subspace and let $\varepsilon$ be a 2-form on $E$, i.e. a bilinear form with $\varepsilon(X,Y)=-\varepsilon(Y,X)$. Then

is a maximal isotropic subspace: it is isotropic for $\langle X+\xi,Y+\eta\rangle=\tfrac12(\xi(Y)+\eta(X))$ and has dimension $N$.

This is the construction direction of the classification of maximal isotropic subspaces.

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.