Can you take nine dyads for the dyad base?

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$\vec{e^k} \otimes \vec{e^l} (k,l = 1,2,3) $

I do not understand how to prove that this can be taken for nine dyads. Is there a version to use $\vec{e_i} \otimes \vec{e^k} (i,k = 1,2,3)$, or the correct course of thought?