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Given a pair <math>(\mathbf{E}, \varepsilon)</math> one can construct a maximally isotropic subbundle <math>L(\mathbf{E}, \varepsilon)</math> of <math>\mathbf{T} \oplus \mathbf{T}^*</math> as follows. The elements of the subbundle are the [[formal sum]]s <math>X+\xi</math> where the [[vector field]] ''X'' is a section of '''E''' and the one-form ''ξ'' restricted to the [[dual space]] <math>\mathbf{E}^*</math> is equal to the one-form <math>\varepsilon(X).</math>
To see that <math>L(\mathbf{E}, \varepsilon)</math> is isotropic, notice that if ''Y'' is a section of '''E''' and <math>\xi</math> restricted to <math>\mathbf{E}^*</math> is <math>\varepsilon(X)</math> then <math>\xi(Y) =\varepsilon(X,Y),</math> as the part of <math>\xi</math> orthogonal to <math>\mathbf{E}^*</math> annihilates ''Y''.
:<math>\langle X+\xi,Y+\eta\rangle=\frac{1}{2}(\xi(Y)+\eta(X))=\frac{1}{2}(\varepsilon(Y,X)+\varepsilon(X,Y))=0</math>
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