# 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 ℝ7 a vector a × b also in ℝ7.[1] Like the cross product in three dimensions the seven-dimensional product is anticommutative and a × b is orthogonal to both a and b. Unlike in three dimensions, it does not satisfy the Jacobi identity. And while the three-dimensional cross product is unique up to a change in sign, there are many seven-dimensional cross products. The seven-dimensional cross product has the same relationship to octonions as the three-dimensional product does to quaternions.

The seven-dimensional cross product is one way of generalising the cross product to other than three dimensions, and it turns out to be the only other non-trivial bilinear product of two vectors that is vector valued, anticommutative and orthogonal.[2] In other dimensions there are vector-valued products of three or more vectors that satisfy these conditions, and binary products with bivector results.

## Example

The postulates underlying construction of the seven-dimensional cross product are presented in the section Definition. As context for that discussion, the historically first example of the cross product is tabulated below using e1 to e7 as basis vectors.[3][4] This table is one of 480 independent multiplication tables fitting the pattern that each unit vector appears once in each column and once in each row.[5] Thus, each unit vector appears as a product in the table six times, three times with a positive sign and three with a negative sign because of antisymmetry about the diagonal of zero entries. For example, e1 = e2 × e3 = e4 × e5 = e7 × e6 and the negative entries are the reversed cross products.

 Number 1 2 3 4 5 6 7 Letter i j k l il jl kl Alternate i j k l m n o
Cayley's sample multiplication table
× e1 e2 e3 e4 e5 e6 e7
e1 0 e3 e2 e5 e4 e7 e6
e2 e3 0 e1 e6 e7 e4 e5
e3 e2 e1 0 e7 e6 e5 e4
e4 e5 e6 e7 0 e1 e2 e3
e5 e4 e7 e6 e1 0 e3 e2
e6 e7 e4 e5 e2 e3 0 e1
e7 e6 e5 e4 e3 e2 e1 0

Entries in the interior give the product of the corresponding vectors on the left and the top in that order (the product is anti-commutative). Some entries are highlighted to emphasize the symmetry.

The table can be summarized by the relation[4]

$\mathbf{e}_i \mathbf{\times} \mathbf{e}_j = \varepsilon _{ijk} \mathbf{e}_k \ ,$

where $\varepsilon _{ijk}$ is a completely antisymmetric tensor with a positive value +1 when ijk = 123, 145, 176, 246, 257, 347, 365. By picking out the factors leading to the unit vector e1, for example, one finds the formula for the e1 component of x × y. Namely

$\left( \mathbf{ x \times y}\right)_1 = x_2y_3 - x_3y_2 +x_4y_5-x_5y_4 + x_7y_6-x_6y_7 = -\left( \mathbf{ y \times x}\right)_1 \ .$

The top left 3 × 3 corner of the table is the same as the cross product in three dimensions. It also may be noticed that orthogonality of the cross product to its constituents x and y is a requirement upon the entries in this table. However, because of the many possible multiplication tables, general results for the cross product are best developed using a basis-independent formulation, as introduced next.

## Definition

We can define a cross product on a Euclidean space V as a bilinear map from V × V to V mapping vectors x and y in V to another vector x × y also in V, where x × y has the properties[1][6]

$\mathbf{x} \cdot (\mathbf{x} \times \mathbf{y}) = (\mathbf{x} \times \mathbf{y}) \cdot \mathbf{y}=0$,
$|\mathbf{x} \times \mathbf{y}|^2 = |\mathbf{x}|^2 |\mathbf{y}|^2 - (\mathbf{x} \cdot \mathbf{y})^2$

where (x·y) is the Euclidean dot product and |x| is the vector norm. The first property states that the cross product is perpendicular to its arguments, while the second property gives the magnitude of the cross product. An equivalent expression in terms of the angle θ between the vectors[7] is[8]

$|\mathbf{x} \times \mathbf{y}| = |\mathbf{x}| |\mathbf{y}| \sin \theta,$

or the area of the parallelogram in the plane of x and y with the two vectors as sides.[9] As a third alternative the following can be shown to be equivalent to either expression for the magnitude:[10]

$|\mathbf{x} \times \mathbf{y}| = |\mathbf{x}| |\mathbf{y}|~\mbox{if} \ \left( \mathbf{x} \cdot \mathbf{y} \right)= 0.$

## 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.[2][8][10] This restriction upon dimensionality 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 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.[11]

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:[8]

“…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:
$\mathbf{x} \times \mathbf{y} = -\mathbf{y} \times \mathbf{x}$,
2. Scalar triple product:
$\mathbf{x} \cdot (\mathbf{y} \times \mathbf{z}) = \mathbf{y} \cdot (\mathbf{z} \times \mathbf{x}) = \mathbf{z} \cdot (\mathbf{x} \times \mathbf{y})$
3. Malcev identity:[8]
$(\mathbf{x} \times \mathbf{y}) \times (\mathbf{x} \times \mathbf{z}) = ((\mathbf{x} \times \mathbf{y}) \times \mathbf{z}) \times \mathbf{x} + ((\mathbf{y} \times \mathbf{z}) \times \mathbf{x}) \times \mathbf{x} + ((\mathbf{z} \times \mathbf{x}) \times \mathbf{x}) \times \mathbf{y}$
$\mathbf{x} \times (\mathbf{x} \times \mathbf{y}) = -|\mathbf{x}|^2 \mathbf{y} + (\mathbf{x} \cdot \mathbf{y}) \mathbf{x}.$

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

1. Vector triple product:
$\mathbf{x} \times (\mathbf{y} \times \mathbf{z}) = (\mathbf{x} \cdot \mathbf{z}) \mathbf{y} - (\mathbf{x} \cdot \mathbf{y}) \mathbf{z}$
2. Jacobi identity:[8]
$\mathbf{x} \times (\mathbf{y} \times \mathbf{z}) + \mathbf{y} \times (\mathbf{z} \times \mathbf{x}) + \mathbf{z} \times (\mathbf{x} \times \mathbf{y}) = 0$

## 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 multiplication table is described in the Example section, but it is not unique.[5] Unlike three dimensions, there are many tables because every pair of unit vectors is perpendicular to five other unit vectors, allowing many choices for each cross product.

Once we have established a multiplication table, it is then applied to general vectors x and y by expressing x and y in terms of the basis and expanding x × y through bilinearity.

× e1 e2 e3 e4 e5 e6 e7
e1 0 e4 e7 e2 e6 e5 e3
e2 e4 0 e5 e1 e3 e7 e6
e3 e7 e5 0 e6 e2 e4 e1
e4 e2 e1 e6 0 e7 e3 e5
e5 e6 e3 e2 e7 0 e1 e4
e6 e5 e7 e4 e3 e1 0 e2
e7 e3 e6 e1 e5 e4 e2 0
Lounesto's multiplication table

Using e1 to e7 for the basis vectors a different multiplication table from the one in the Introduction, leading to a different cross product, is given with anticommutativity by[8]

$\mathbf{e}_1 \times \mathbf{e}_2 = \mathbf{e}_4, \quad \mathbf{e}_2 \times \mathbf{e}_4 = \mathbf{e}_1, \quad \mathbf{e}_4 \times \mathbf{e}_1 = \mathbf{e}_2,$
$\mathbf{e}_2 \times \mathbf{e}_3 = \mathbf{e}_5, \quad \mathbf{e}_3 \times \mathbf{e}_5 = \mathbf{e}_2, \quad \mathbf{e}_5 \times \mathbf{e}_2 = \mathbf{e}_3,$
$\mathbf{e}_3 \times \mathbf{e}_4 = \mathbf{e}_6, \quad \mathbf{e}_4 \times \mathbf{e}_6 = \mathbf{e}_3, \quad \mathbf{e}_6 \times \mathbf{e}_3 = \mathbf{e}_4,$
$\mathbf{e}_4 \times \mathbf{e}_5 = \mathbf{e}_7, \quad \mathbf{e}_5 \times \mathbf{e}_7 = \mathbf{e}_4, \quad \mathbf{e}_7 \times \mathbf{e}_4 = \mathbf{e}_5,$
$\mathbf{e}_5 \times \mathbf{e}_6 = \mathbf{e}_1, \quad \mathbf{e}_6 \times \mathbf{e}_1 = \mathbf{e}_5, \quad \mathbf{e}_1 \times \mathbf{e}_5 = \mathbf{e}_6,$
$\mathbf{e}_6 \times \mathbf{e}_7 = \mathbf{e}_2, \quad \mathbf{e}_7 \times \mathbf{e}_2 = \mathbf{e}_6, \quad \mathbf{e}_2 \times \mathbf{e}_6 = \mathbf{e}_7,$
$\mathbf{e}_7 \times \mathbf{e}_1 = \mathbf{e}_3, \quad \mathbf{e}_1 \times \mathbf{e}_3 = \mathbf{e}_7, \quad \mathbf{e}_3 \times \mathbf{e}_7 = \mathbf{e}_1.$

More compactly this rule can be written as

$\mathbf{e}_i \times \mathbf{e}_{i+1} = \mathbf{e}_{i+3}$

with i = 1...7 modulo 7 and the indices i, i + 1 and i + 3 allowed to permute evenly. Together with anticommutativity this generates the product. This rule directly produces the two diagonals immediately adjacent to the diagonal of zeros in the table. Also, from an identity in the subsection on consequences,

$\mathbf{e}_i \times \left( \mathbf{e}_i \times \mathbf{e}_{i+1}\right) =-\mathbf{e}_{i+1} = \mathbf{e}_i \times \mathbf{e}_{i+3} \ ,$

which produces diagonals further out, and so on.

The ej component of cross product x × y is given by selecting all occurrences of ej in the table and collecting the corresponding components of x from the left column and of y from the top row. The result is:

\begin{align}\mathbf{x} \times \mathbf{y} = (x_2y_4 - x_4y_2 + x_3y_7 - x_7y_3 + x_5y_6 - x_6y_5)\,&\mathbf{e}_1 \\ {}+ (x_3y_5 - x_5y_3 + x_4y_1 - x_1y_4 + x_6y_7 - x_7y_6)\,&\mathbf {e}_2 \\ {}+ (x_4y_6 - x_6y_4 + x_5y_2 - x_2y_5 + x_7y_1 - x_1y_7)\,&\mathbf{e}_3 \\ {}+ (x_5y_7 - x_7y_5 + x_6y_3 - x_3y_6 + x_1y_2 - x_2y_1)\,&\mathbf{e}_4 \\ {}+ (x_6y_1 - x_1y_6 + x_7y_4 - x_4y_7 + x_2y_3 - x_3y_2)\,&\mathbf{e}_5 \\ {}+ (x_7y_2 - x_2y_7 + x_1y_5 - x_5y_1 + x_3y_4 - x_4y_3)\,&\mathbf{e}_6 \\ {}+ (x_1y_3 - x_3y_1 + x_2y_6 - x_6y_2 + x_4y_5 - x_5y_4)\,&\mathbf{e}_7. \\ \end{align}

As the cross product is bilinear the operator x×– can be written as a matrix, which takes the form[citation needed]

$T_{\mathbf x} = \begin{bmatrix} 0 & -x_4 & -x_7 & x_2 & -x_6 & x_5 & x_3 \\ x_4 & 0 & -x_5 & -x_1 & x_3 & -x_7 & x_6 \\ x_7 & x_5 & 0 & -x_6 & -x_2 & x_4 & -x_1 \\ -x_2 & x_1 & x_6 & 0 & -x_7 & -x_3 & x_5 \\ x_6 & -x_3 & x_2 & x_7 & 0 & -x_1 & -x_4 \\ -x_5 & x_7 & -x_4 & x_3 & x_1 & 0 & -x_2 \\ -x_3 & -x_6 & x_1 & -x_5 & x_4 & x_2 & 0 \end{bmatrix}.$

The cross product is then given by

$\mathbf{x} \times \mathbf{y} = T_{\mathbf{x}}(\mathbf{y}).$

### Different multiplication tables

Fano planes for the two multiplication tables used here.

Two different multiplication tables have been used in this article, and there are more.[5][12] These multiplication tables are characterized by the Fano plane,[13][14] and these are shown in the figure for the two tables used here: at top, the one described by Sabinin, Sbitneva, and Shestakov, and at bottom that described by Lounesto. The numbers under the Fano diagrams (the set of lines in the diagram) indicate a set of indices for seven independent products in each case, interpreted as ijkei × ej = ek. The multiplication table is recovered from the Fano diagram by following either the straight line connecting any three points, or the circle in the center, with a sign as given by the arrows. For example, the first row of multiplications resulting in e1 in the above listing is obtained by following the three paths connected to e1 in the lower Fano diagram: the circular path e2 × e4, the diagonal path e3 × e7, and the edge path e6 × e1 = e5 rearranged using one of the above identities as:

$\mathbf{e_6 \times} \left( \mathbf{e_6 \times e_1} \right) = -\mathbf{e_1} = \mathbf {e_6 \times e_5} ,$

or

$\mathbf {e_5 \times e_6} =\mathbf{e_1} ,$

also obtained directly from the diagram with the rule that any two unit vectors on a straight line are connected by multiplication to the third unit vector on that straight line with signs according to the arrows (sign of the permutation that orders the unit vectors).

It can be seen that both multiplication rules follow from the same Fano diagram by simply renaming the unit vectors, and changing the sense of the center unit vector. The question arises: how many multiplication tables are there?[14]

The question of possible multiplication tables arises, for example, when one reads another article on octonions, which uses a different one from the one given by [Cayley, say]. Usually it is remarked that all 480 possible ones are equivalent, that is, given an octonionic algebra with a multiplication table and any other valid multiplication table, one can choose a basis such that the multiplication follows the new table in this basis. One may also take the point of view, that there exist different octonionic algebras, that is, algebras with different multiplication tables. With this interpretation...all these octonionic algebras are isomorphic.

—Jörg Schray, Corinne A Manogue, Octonionic representations of Clifford algebras and triality (1994)

### Using geometric algebra

The product can also be calculated using geometric algebra. The product starts with the exterior product, a bivector valued product of two vectors:

$\mathbf{B} = \mathbf{x} \wedge \mathbf{y} = \frac{1}{2}(\mathbf{xy} - \mathbf{yx}).$

This is bilinear, alternate, has the desired magnitude, but is not vector valued. The vector, and so the cross product, comes from the product of this bivector with a trivector. In three dimensions up to a scale factor there is only one trivector, the pseudoscalar of the space, and a product of the above bivector and one of the two unit trivectors gives the vector result, the dual of the bivector.

A similar calculation is done is seven dimensions, except as trivectors form a 35-dimensional space there are many trivectors that could be used, though not just any trivector will do. The trivector that gives the same product as the above coordinate transform is

$\mathbf{v} = \mathbf{e}_{124} + \mathbf{e}_{235} + \mathbf{e}_{346} + \mathbf{e}_{457} + \mathbf{e}_{561} + \mathbf{e}_{672} + \mathbf{e}_{713}.$

This is combined with the exterior product to give the cross product

$\mathbf{x} \times \mathbf{y} = -(\mathbf{x} \wedge \mathbf{y}) ~\lrcorner~ \mathbf{v}$

where $\lrcorner$ is the left contraction operator from geometric algebra.[8][15]

## Relation to the octonions

Just as the 3-dimensional cross product can be expressed in terms of the quaternions, the 7-dimensional cross product can be expressed in terms of the octonions. After identifying ℝ7 with the imaginary octonions (the orthogonal complement of the real line in O), the cross product is given in terms of octonion multiplication by

$\mathbf x \times \mathbf y = \mathrm{Im}(\mathbf{xy}) = \frac{1}{2}(\mathbf{xy}-\mathbf{yx}).$

Conversely, suppose V is a 7-dimensional Euclidean space with a given cross product. Then one can define a bilinear multiplication on ℝ⊕V as follows:

$(a,\mathbf{x})(b,\mathbf{y}) = (ab - \mathbf{x}\cdot\mathbf{y}, a\mathbf y + b\mathbf x + \mathbf{x}\times\mathbf{y}).$

The space ℝ⊕V with this multiplication is then isomorphic to the octonions.[16]

The cross product only exists in three and seven dimensions as one can always define a multiplication on a space of one higher dimension as above, and this space can be shown to be a normed division algebra. By Hurwitz's theorem such algebras only exist in one, two, four, and eight dimensions, so the cross product must be in zero, one, three or seven dimensions. The products in zero and one dimensions are trivial, so non-trivial cross products only exist in three and seven dimensions.[17][18]

The failure of the 7-dimension cross product to satisfy the Jacobi identity is due to the nonassociativity of the octonions. In fact,

$\mathbf{x}\times(\mathbf{y}\times\mathbf{z}) + \mathbf{y}\times(\mathbf{z}\times\mathbf{x}) + \mathbf{z}\times(\mathbf{x}\times\mathbf{y}) = -\frac{3}{2}[\mathbf x, \mathbf y, \mathbf z]$

where [x, y, z] is the associator.

## Rotations

In three dimensions the cross product is invariant under the group of the rotation group, SO(3), so the cross product of x and y after they are rotated is the image of x × y under the rotation. But this invariance is not true in seven dimensions; that is, the cross product is not invariant under the group of rotations in seven dimensions, SO(7). Instead it is invariant under the exceptional Lie group G2, a subgroup of SO(7).[8][16]

## Generalizations

Non-trivial binary cross products exist only in three and seven dimensions. But if the restriction that the product is binary is lifted, so products of more than two vectors are allowed, then more products are possible.[19][20] As in two dimensions the product must be vector valued, linear, and anti-commutative in any two of the vectors in the product.

The product should satisfy orthogonality, so it is orthogonal to all its members. This means no more than n − 1 vectors can be used in n dimensions. The magnitude of the product should equal the volume of the parallelotope with the vectors as edges, which is can be calculated using the Gram determinant. So the conditions are

• orthogonality:
$\left( \mathbf{a_1} \times \ \cdots \ \times \mathbf{a_k}\right) \cdot \mathbf{a_j} = 0$
• the Gram determinant:
$|\mathbf{a_1} \times \ \cdots \ \times \mathbf{a_k} |^2 = \det (\mathbf{a_i \cdot a_j}) = \begin{vmatrix} \mathbf {a_1 \cdot a_1} & \mathbf {a_1 \cdot a_2} & \dots & \mathbf {a_1 \cdot a_k}\\ \mathbf {a_2 \cdot a_1} & \mathbf {a_2 \cdot a_2} & \dots & \mathbf {a_2 \cdot a_k}\\ \dots & \dots & \dots & \dots\\ \mathbf {a_k \cdot a_1} & \mathbf {a_k \cdot a_2} & \dots & \mathbf {a_k \cdot a_k}\\ \end{vmatrix}$

The Gram determinant is the squared volume of the parallelotope with a1, ..., ak as edges. If there are just two vectors x and y it simplifies to the condition for the binary cross product given above, that is

$|\mathbf{x} \times \mathbf{y}|^2 = \begin{vmatrix} \mathbf {x \cdot x} & \mathbf {x \cdot y}\\ \mathbf {y \cdot x} & \mathbf {y \cdot y}\\ \end{vmatrix} = |\mathbf{x}|^2 |\mathbf{y}|^2 - (\mathbf{x} \cdot \mathbf{y})^2 ,$

With these conditions a non-trivial cross product only exists:

• as a binary product in three and seven dimensions
• as a product of n − 1 vectors in n > 3 dimensions
• as a product of three vectors in eight dimensions

The product of n − 1 vectors is in n dimensions is the Hodge dual of the exterior product of n − 1 vectors. One version of the product of three vectors in eight dimensions is given by

$\mathbf{a} \times \mathbf{b} \times \mathbf{c} = (\mathbf{a} \wedge \mathbf{b} \wedge \mathbf{c}) ~\lrcorner~ (\mathbf{w} - \mathbf{ve}_8)$

where v is the same trivector as used in seven dimensions, $\lrcorner$ is again the left contraction, and w = −ve12...7 is a 4-vector.

## Notes

1. ^ a b WS Massey (1983). "Cross products of vectors in higher dimensional Euclidean spaces". The American Mathematical Monthly (Mathematical Association of America) 90 (10): 697–701. doi:10.2307/2323537. JSTOR 2323537.
2. ^ a b WS Massey (1983). "Cross products of vectors in higher dimensional Euclidean spaces". The American Mathematical Monthly 90 (10): 697–701. doi:10.2307/2323537. JSTOR 2323537. "If one requires only three basic properties of the cross product ... it turns out that a cross product of vectors exists only in 3-dimensional and 7-dimensional Euclidean space."
3. ^ This table is due to Arthur Cayley (1845) and John T. Graves (1843). See G Gentili, C Stoppato, DC Struppa and F Vlacci (2009). "Recent developments for regular functions of a hypercomplex variable". In Irene Sabadini, M Shapiro, F Sommen. Hypercomplex analysis (Conference on quaternionic and Clifford analysis; proceedings ed.). Birkaüser. p. 168. ISBN 978-3-7643-9892-7.
4. ^ a b Lev Vasilʹevitch Sabinin, Larissa Sbitneva, I. P. Shestakov (2006). "§17.2 Octonion algebra and its regular bimodule representation". Non-associative algebra and its applications. CRC Press. p. 235. ISBN 0-8247-2669-3
5. ^ a b c Rafał Abłamowicz, Pertti Lounesto, Josep M. Parra (1996). "§ Four ocotonionic basis numberings". Clifford algebras with numeric and symbolic computations. Birkhäuser. p. 202. ISBN 0-8176-3907-1.
6. ^ Mappings are restricted to be bilinear by (Massey 1993) and Robert B Brown and Alfred Gray (1967). "Vector cross products". Commentarii Mathematici Helvetici (Birkhäuser Basel) 42 (1/December): 222–236. doi:10.1007/BF02564418..
7. ^ The definition of angle in n-dimensions ordinarily is defined in terms of the dot product as:
$(\mathbf{x \cdot y}) = |\mathbf x ||\mathbf y | \ \cos \theta \ , \ \mathrm {in \ the \ range } \ (-\pi < \theta \le \pi )\ ,$
where θ is the angle between the vectors. Consequently, this property of the cross product provides its magnitude as:
$|\mathbf{ x \times y} |^2 =|\mathbf x |^2 |\mathbf y |^2 \left(1 - \cos^2 \theta \right) \ .$
From the Pythagorean trigonometric identity this magnitude equals
$|\mathbf{x} \times \mathbf{y}| = |\mathbf{x}| |\mathbf{y}| \sin \theta$.
See Francis Begnaud Hildebrand (1992). Methods of applied mathematics (Reprint of Prentice-Hall 1965 2nd ed.). Courier Dover Publications. p. 24. ISBN 0-486-67002-3.
8. Lounesto, pp. 96–97
9. ^ Kendall, M. G. (2004). A Course in the Geometry of N Dimensions. Courier Dover Publications. p. 19. ISBN 0-486-43927-5.
10. ^ a b Z.K. Silagadze (2002). "Multi-dimensional vector product". Journal of Physics A: Mathematical and General 35 (23): 4949. arXiv:math.RA/0204357. doi:10.1088/0305-4470/35/23/310.
11. ^ Nathan Jacobson (2009). Basic algebra I (Reprint of Freeman 1974 2nd ed.). Dover Publications. pp. 417–427. ISBN 0-486-47189-6.
12. ^ Further discussion of the tables and the connection of the Fano plane to these tables is found here: Tony Smith. "Octonion products and lattices". Retrieved 2010-07-11.
13. ^ Rafał Abłamowicz, Bertfried Fauser (2000). Clifford Algebras and Their Applications in Mathematical Physics: Algebra and physics. Springer. p. 26. ISBN 0-8176-4182-3.
14. ^ a b Jörg Schray, Corinne A. Manogue (1996). "Octonionic representations of Clifford algebras and triality". Foundations of physics (Springer) 26 (1/January): 17–70. doi:10.1007/BF02058887. Available as ArXive preprint Figure 1 is located here.
15. ^ Bertfried Fauser (2004). "§18.4.2 Contractions". In Pertti Lounesto, Rafał Abłamowicz. Clifford algebras: applications to mathematics, physics, and engineering. Birkhäuser. pp. 292 ff. ISBN 0-8176-3525-4.
16. ^ a b John C. Baez (2001). "The Octonions". Bull. Amer. Math. 39: 38.
17. ^ Elduque, Alberto (2004). Vector cross products.
18. ^ Darpö, Erik (2009). "Vector product algebras". Bulletin of the London Mathematical Society 41 (5): 898–902. doi:10.1112/blms/bdp066. See also: Real vector product algebras. CiteSeerX: 10.1.1.66.4.
19. ^ Lounesto, §7.5: Cross products of k vectors in ℝn, p. 98
20. ^ Jean H. Gallier (2001). "Problem 7.10 (2)". Geometric methods and applications: for computer science and engineering. Springer. p. 244. ISBN 0-387-95044-3.