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 ⁠${\displaystyle \mathbb {R} ^{7}}$⁠.^[1] 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 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.^[2] In other dimensions there are vector-valued products of three or more vectors that satisfy these conditions, and binary products with bivector results.