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Let $V$ be a real $n$-dimensional vector space, and let $W \le \bigwedge^k V$ be a subspace . Suppose that $\dim W \ge 2$. Does $W$ contain a non-zero decomposable element?

If $\dim W=1$, then clearly we can take $W=\text{span} (\sigma)$ for some non-decomposable $\sigma \in \bigwedge^k V$.

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