angle θ between the vectors is which is the area of the parallelogram in the plane of x and y with the two vectors as sides. A third statement of the magnitude condition is if x × x = 0 is assumed as a separate axiom. Consequences of the defining properties Given the properties of bilinearity, orthogonality and magnitude, a nonzero cross product exists only in three and seven dimensions. This can be shown by postulating the properties required for the cross product, then deducing an 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 the 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 formed from the product of the normed division algebra by restricting it to the 0, 1, 3, or 7 imaginary dimensions of the algebra, giving nonzero products in only three and seven dimensions. In contrast to the three-dimensional cross product, which 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 and any vector v of magnitude |v| = |x||y| sin θ 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. Unlike in three dimensions, x × y = a × b does not imply that a and b lie in the same plane as x and y. Further properties follow from the definition, including the following identities: Anticommutativity: Scalar triple product: Malcev identity: Other properties follow only in the three-dimensional case, and are not satisfied by the seven-dimensional cross product, notably, Vector triple product: Jacobi identity: Because the Jacobi Identity is not satisfied, the seven-dimensional cross product does not give R7 the structure of a Lie algebra. Coordinate expressions To define a particular cross product, an orthonormal basis {ej} may be selected and a multiplication table provided that determines all the products {ei × ej}. One possible