Seven dimensional cross product

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In mathematics, the seven dimensional cross product is a generalization of the three dimensional cross product. This is obtained by using the octonions in place of the quaternion notation for the 3D cross product.

This seven-dimensional cross product is a nonassociative algebra over the real numbers. It has the following properties in common with the usual three-dimensional cross product:

It is bilinear in the sense that

x × (ay + bz) = ax × y + bx × z

(ay + bz) × x = ay × x + bz × x.

It is anticommutative:

x × y + y × x = 0

It is perpendicular to both x and y:

x · (x × y) = y · (x × y) = 0

We have

|x × y|² = |x|² |y|² − (x · y)².

Unlike the three-dimensional cross product, it does not however satisfy the Jacobi identity (equality holds for the three dimensional cross product):

x × (y × z) + y × (z × x) + z × (x × y) ≠ 0

so the seven dimensional cross product is not a Lie algebra.

The cross product can not be extended to a dimension higher than 7, since the cross products are closely linked to composition algebras, of which the octonions are the largest.