Patrick D. Baier in his Ph.D. thesis for proving the theorem 2.1.4 used the following non-trivial fact (in chapter 2 on page 14):

Let $0\neq X\in V$ (here $V$ is of dimension 6), $W^\ast = Ann(X)$ and $\Omega\in\wedge^3 V^\ast$. Then we can find unique elements $\psi\in\wedge^2 V^\ast$ and $\phi\in\wedge^3 V^\ast$ such that $\Omega=\theta\wedge\psi + \phi$ (I think this decomposition is from Hitchin in his paper about geometry of three forms in six dimensions) so that $i_X\Omega=\psi$ (why?). Moreover, there are unique elements $r\in\mathbb{R}$ and $Y\in Ker(\theta)$ (here $\theta\in V^\ast$ with $\theta (X)=1$) such that (why?) we have

1) $i_{(rX)} v = \psi\wedge\phi$

2) $i_Y v_0 = \psi\wedge\psi$

(here $v_0\in\wedge^5 W^\ast$ is the unique element such that $v=\theta\wedge v_0$)

`$v \in \Lambda^6 V^*$`

and`$\theta \in V^*$`

.`$i_X v$`

and`$\psi \wedge \phi$`

are both in`$\ker\; i_X \cong \Lambda^5 W^* \subset \Lambda^5 V^*$`

, so must be proportional.`$\ker \theta \to \Lambda^4 W^*, \; Y \mapsto i_Y v_0$`

is an isomorphism. – Johannes Nordström Feb 3 '13 at 10:12