The 4D cross product A×B, defined by the fourth quadrant of the octonion multiplication table, is Im(ĀB) in quaternion form. It is invariant under left multiplication by unit quaternions and covariant, as a 3D rotation, under right multiplication. The second cross product (A×B)′ = Im(AB̄) is, conversely, right-invariant and left-covariant. From this left invariance the following results follow at once.
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Sung-tae Lim (2026) studied this question.
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