Seven-dimensional cross product

In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in seven-dimensional Euclidean space. It assigns to any two vectors a, b in ⁠${\displaystyle \mathbb {R} ^{7}}$⁠ a vector a × b also in ⁠${\displaystyle \mathbb {R} ^{7}}$⁠. Like the cross product in three dimensions, the seven-dimensional product is anticommutative and a × b is orthogonal both to a and to b. Unlike in three dimensions, it does not satisfy the Jacobi identity, and while the three-dimensional cross product is unique up to a sign, there are many seven-dimensional cross products. The seven-dimensional cross product has the same relationship to the octonions as the three-dimensional product does to the quaternions.

The seven-dimensional cross product is one way of generalizing the cross product to dimensions other than three dimensions, and it is the only other bilinear product of two vectors that is vector-valued, orthogonal, and has the same magnitude as in the 3D case. In other dimensions there are vector-valued products of three or more vectors that satisfy these conditions, and binary products with bivector results.