to be a normed division algebra. By Hurwitz's theorem such algebras only exist in one, two, four, and eight dimensions, so the cross product must be in zero, one, three or seven dimensions. The products in zero and one dimensions are trivial, so non-trivial cross products only exist in three and seven dimensions. The failure of the 7-dimension cross product to satisfy the Jacobi identity is due to the nonassociativity of the octonions. In fact, where [x, y, z] is the associator. Rotations In three dimensions the cross product is invariant under the action of the rotation group, SO(3), so the cross product of x and y after they are rotated is the image of under the rotation. But this invariance is not true in seven dimensions; that is, the cross product is not invariant under the group of rotations in seven dimensions, SO(7). Instead it is invariant under the exceptional Lie group G2, a subgroup of SO(7). Generalizations Nonzero binary cross products exist only in three and seven dimensions. Further products are possible when lifting the restriction that it must be a binary product. We require the product to be multi-linear, alternating, vector-valued, and orthogonal to each of the input vectors ai. The orthogonality requirement implies that in n dimensions, no more than vectors can be used. The magnitude of the product should equal the volume of the parallelotope with the vectors as edges, which can be calculated using the Gram determinant. The conditions are orthogonality: for . the Gram determinant: The Gram determinant is the squared volume of the parallelotope with a1, ..., ak as edges. With these conditions a non-trivial cross product only exists: as a binary product in three and seven dimensions as a product of n − 1 vectors in n ≥ 3 dimensions, being the Hodge dual of the exterior product of the vectors as a product of three vectors in eight dimensions One version of the product of three vectors in eight dimensions is given by where v is the same trivector as used in seven dimensions, is again the left contraction, and is a 4-vector. There are also trivial products. As noted already, a binary product only exists in 7, 3, 1 and 0 dimensions, the last two being identically zero. A further trivial 'product' arises in even dimensions, which takes a single vector and produces a vector of the same magnitude