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Let $V$ be be a vector space of dimension $m$ over any field and $\ell\leq m$ be a positive integer. Let $\omega_1,\ldots,\omega_r \in\bigwedge^\ell V$ are linearly independent, completely decomposable vectors such that their sum $\omega=\omega_1+\cdots+\omega_r$ is again completely decomposable. Is it true then that, $\omega´=\omega_1+\cdots+\omega_j$ for any $j\leq r$ is completely decomposable?

This might be a simple problem. I was trying to prove this but neither can prove nor can produce a counterexample. Any help or reference would be appreciatable.

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No. Take $l = 2$, $m = 4$, $\omega_1 = 2e_1 \wedge e_3$, $\omega_2 = 2e_2 \wedge e_4$, $\omega_3 = (-e_1 + e_2) \wedge (e_3 - e_4)$.

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