Seven-dimensional Cross Product - Consequences of The Defining Properties

Consequences of The Defining Properties

Given the three basic properties of (i) bilinearity, (ii) orthogonality and (iii) magnitude discussed in the section on definition, a nontrivial cross product exists only in three and seven dimensions. This restriction upon dimensionality can be shown by postulating the properties required for the cross product, then deducing a equation which is only satisfied when the dimension is 0, 1, 3 or 7. In zero dimensions there is only the zero vector, while in one dimension all vectors are parallel, so in both these cases a cross product must be identically zero.

The restriction to 0, 1, 3 and 7 dimensions is related to Hurwitz's theorem, that normed division algebras are only possible in 1, 2, 4 and 8 dimensions. The cross product is derived from the product of the algebra by considering the product restricted to the 0, 1, 3, or 7 imaginary dimensions of the algebra. Again discarding trivial products the product can only be defined this way in three and seven dimensions.

In contrast with three dimensions where the cross product is unique (apart from sign), there are many possible binary cross products in seven dimensions. One way to see this is to note that given any pair of vectors x and y ∈ ℝ7 and any vector v of magnitude |v| = |x||y| sinθ in the five dimensional space perpendicular to the plane spanned by x and y, it is possible to find a cross product with a multiplication table (and an associated set of basis vectors) such that x × y = v. That leaves open the question of just how many vector pairs like x and y can be matched to specified directions like v before the limitations of any particular table intervene.

Another difference between the three dimensional cross product and a seven dimensional cross product is:

“…for the cross product x × y in ℝ7 there are also other planes than the linear span of x and y giving the same direction as x × y” —Pertti Lounesto, Clifford algebras and spinors, p. 97

This statement is exemplified by every multiplication table, because any specific unit vector selected as a product occurs as a mapping from three different pairs of unit vectors, once with a plus sign and once with a minus sign. Each of these different pairs, of course, corresponds to another plane being mapped into the same direction.

Further properties follow from the definition, including the following identities:

1. Anticommutativity:

   ,

2. Scalar triple product:

3. Malcev identity:

Other properties follow only in the three dimensional case, and are not satisfied by the seven dimensional cross product, notably,

1. Vector triple product:

2. Jacobi identity: