Unlike the 3-dimensional cross product, the 7-dimension cross product is not unique (up to a sign), since any of the seven unit vectors can be the product of three distinct pairs from among the other six.
Unlike the 3-dimensional cross product, the 7-dimension cross product is not unique (up to a sign), because the same cross product can result from multiplying vectors in different planes. In particular, any one of the seven unit vectors can be the product of three distinct pairs from among the other six.