Let $a, b, c, d$ be vectors with integer coordinates in $\mathbb{R}^4$ such that $k a \wedge b = c \wedge d$ for some integer $k$ and $a \wedge b \neq l v$ for any $v \in \bigwedge^2 (\mathbb{R}^4)$ with integer coefficients and integer $l \neq \pm 1$. Prove that $c$ and $d$ can expressed as linear combinations of $a$ and $b$ with integer coefficients.
If anybody has any suggestions on which gcds should be checked or on the problem in general, you have my gratitude.