In summary: \epsilon_{ijk}=\sum_{i=0}^3a_ib_j(\mathbf{e_k}-\mathbf{e_1})\mathbf{e_2}-\sum_{i=0}^3a_ib_j(\mathbf{e_k}+\mathbf{e_1})-\sum_{i=0}^3a_1b_3\mathbf{e_2}+\sum_{i=0}^3a_2b_1\mathbf{e_3}\end{align*}...the i-th component of the cross product.
#36
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ahh so its just the identity dij? so the answer is aibi?