Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used. Octonions are related to exceptional structures in mathematics, among them the exceptional Lie groups. Octonions have applications in fields such as string theory, special relativity and quantum logic. Applying the Cayley–Dickson construction to the octonions produces the sedenions.

The octonions were discovered in 1843 by John T. Graves, inspired by his friend William Rowan Hamilton's discovery of quaternions. Graves called his discovery "octaves", and mentioned them in a letter to Hamilton dated 26 December 1843.[1] He first published his result slightly later than Arthur Cayley's article.[2] The octonions were discovered independently by Cayley[3] and are sometimes referred to as "Cayley numbers" or the "Cayley algebra". Hamilton described the early history of Graves' discovery.[4]

The octonions can be thought of as octets (or 8-tuples) of real numbers. Every octonion is a real linear combination of the unit octonions:

where e0 is the scalar or real element; it may be identified with the real number 1. That is, every octonion x can be written in the form

Addition and subtraction of octonions is done by adding and subtracting corresponding terms and hence their coefficients, like quaternions. Multiplication is more complex. Multiplication is distributive over addition, so the product of two octonions can be calculated by summing the products of all the terms, again like quaternions. The product of each pair of terms can be given by multiplication of the coefficients and a multiplication table of the unit octonions, like this one (due to Cayley, 1845, and Graves, 1843):[5]

Most off-diagonal elements of the table are antisymmetric, making it almost a skew-symmetric matrix except for the elements on the main diagonal, as well as the row and column for which e0 is an operand.

where δij is the Kronecker delta (equal to 1 if and only if i = j), and εijk is a completely antisymmetric tensor with value 1 when ijk = 123, 145, 176, 246, 257, 347, 365.

Each of these 480 definitions is invariant up to signs under some 7-cycle of the points (1234567), and for each 7-cycle there are four definitions, differing by signs and reversal of order. A common choice is to use the definition invariant under the 7-cycle (1234567) with e1e2 = e4 — by using the triangular multiplication diagram, or Fano plane below that also shows the sorted list of 124 based 7-cycle triads and its associated multiplication matrices in both en and IJKL format.

A variation of this sometimes used is to label the elements of the basis by the elements , 0, 1, 2, ..., 6, of the projective line over the finite field of order 7. The multiplication is then given by e = 1 and e1e2 = e4, and all expressions obtained from this by adding a constant (modulo 7) to all subscripts: in other words using the seven triples (124) (235) (346) (450) (561) (602) (013). These are the nonzero codewords of the quadratic residue code of length 7 over the Galois field of two elements, GF(2). There is a symmetry of order 7 given by adding a constant mod 7 to all subscripts, and also a symmetry of order 3 given by multiplying all subscripts by one of the quadratic residues 1, 2, 4 mod 7.[7][8]

The multiplication table for a geometric algebra of signature (−−−−) can be given in terms of the following 7 quaternionic triples (omitting the identity element):

(I, j, k), (i, J, k), (i, j, K), (I, J, K), (∗I, i, m), (∗J, j, m), (∗K, k, m)

in which the lowercase items are vectors and the uppercase ones are bivectors and ∗ = mijk (which is the Hodge star operator). If the is forced to be equal to the identity then the multiplication ceases to be associative, but the may be removed from the multiplication table resulting in an octonion multiplication table.

In keeping ∗ = mijk associative and thus not reducing the 4-dimensional geometric algebra to an octonion one, the whole multiplication table can be derived from the equation for . Consider the gamma matrices. The formula defining the fifth gamma matrix shows that it is the of a four-dimensional geometric algebra of the gamma matrices.

A more systematic way of defining the octonions is via the Cayley–Dickson construction. Just as quaternions can be defined as pairs of complex numbers, the octonions can be defined as pairs of quaternions. Addition is defined pairwise. The product of two pairs of quaternions (a, b) and (c, d) is defined by

where z* denotes the conjugate of the quaternion z. This definition is equivalent to the one given above when the eight unit octonions are identified with the pairs

A convenient mnemonic for remembering the products of unit octonions is given by the diagram, which represents the multiplication table of Cayley and Graves.[5][10] This diagram with seven points and seven lines (the circle through 1, 2, and 3 is considered a line) is called the Fano plane. The lines are directional. The seven points correspond to the seven standard basis elements of Im(O) (see definition below). Each pair of distinct points lies on a unique line and each line runs through exactly three points.

Let (a, b, c) be an ordered triple of points lying on a given line with the order specified by the direction of the arrow. Then multiplication is given by

completely defines the multiplicative structure of the octonions. Each of the seven lines generates a subalgebra of O isomorphic to the quaternions H.

Conjugation is an involution of O and satisfies (xy)* = y*x* (note the change in order).

The set of all purely imaginary octonions span a 7-dimensional subspace of O, denoted Im(O).

The product of an octonion with its conjugate, x*x = xx*, is always a nonnegative real number:

This norm agrees with the standard 8-dimensional Euclidean norm on R8.

The existence of a norm on O implies the existence of inverses for every nonzero element of O. The inverse of x ≠ 0, which is the unique octonion x−1 satisfying xx−1 = x−1x = 1, is given by

The octonions do retain one important property shared by R, C, and H: the norm on O satisfies

This equation means that the octonions form a composition algebra. The higher-dimensional algebras defined by the Cayley–Dickson construction (starting with the sedenions) all fail to satisfy this property. They all have zero divisors.

Wider number systems exist which have a multiplicative modulus (for example, 16-dimensional conic sedenions). Their modulus is defined differently from their norm, and they also contain zero divisors.

As shown by Hurwitz, R, C, H, and O are the only normed division algebras over the real numbers. These four algebras also form the only alternative, finite-dimensional division algebras over the real numbers (up to isomorphism).

Not being associative, the nonzero elements of O do not form a group. They do, however, form a loop, specifically a Moufang loop.

This is antisymmetric and imaginary. If it is considered only as a product on the imaginary subspace Im(O) it defines a product on that space, the seven-dimensional cross product, given by

Like the cross product in three dimensions this is a vector orthogonal to x and y with magnitude

But like the octonion product it is not uniquely defined. Instead there are many different cross products, each one dependent on the choice of octonion product.[11]

An automorphism, A, of the octonions is an invertible linear transformation of O which satisfies

The set of all automorphisms of O forms a group called G2.[12] The group G2 is a simply connected, compact, real Lie group of dimension 14. This group is the smallest of the exceptional Lie groups and is isomorphic to the subgroup of Spin(7) that preserves any chosen particular vector in its 8-dimensional real spinor representation. The group Spin(7) is in turn a subgroup of the group of isotopies described below.

An isotopy of an algebra is a triple of bijective linear maps a, b, c such that if xy = z then a(x)b(y) = c(z). For a = b = c this is the same as an automorphism. The isotopy group of an algebra is the group of all isotopies, which contains the group of automorphisms as a subgroup.

The isotopy group of the octonions is the group Spin8(R), with a, b, c acting as the three 8-dimensional representations.[13] The subgroup of elements where c fixes the identity is the subgroup Spin7(R), and the subgroup where a, b, c all fix the identity is the automorphism group G2.

The octonions play a significant role in the classification and construction of other mathematical entities. For example, the exceptional Lie group G2 is the automorphism group of the octonions, and the other exceptional Lie groups F4, E6, E7 and E8 can be understood as the isometries of certain projective planes defined using the octonions.[14] The set of self-adjoint 3 × 3 octonionic matrices, equipped with a symmetrized matrix product, defines the Albert algebra. In discrete mathematics, the octonions provide an elementary derivation of the Leech lattice, and thus they are closely related to the sporadic simple groups.[15][16]

Applications of the octonions to physics have largely been conjectural. For example, in the 1970s, attempts were made to understand quarks by way of an octonionic Hilbert space.[17] It is known that the octonions, and the fact that only four normed division algebras can exist, relates to the spacetime dimensions in which supersymmetric quantum field theories can be constructed.[18][19] Also, attempts have been made to obtain the Standard Model of elementary particle physics from octonionic constructions, for example using the "Dixon algebra" CHO.[20][21]

Octonions have also arisen in the study of black hole entropy and quantum information science.[22][23]

Octonions have been used in solutions to the hand eye calibration problem in robotics.[24]

Deep octonion networks provide a means of efficient and compact expression in machine learning applications.[25]

There are several natural ways to choose an integral form of the octonions. The simplest is just to take the octonions whose coordinates are integers. This gives a nonassociative algebra over the integers called the Gravesian octonions. However it is not a maximal order (in the sense of ring theory); there are exactly seven maximal orders containing it. These seven maximal orders are all equivalent under automorphisms. The phrase "integral octonions" usually refers to a fixed choice of one of these seven orders.

These maximal orders were constructed by Kirmse (1925), Dickson and Bruck as follows. Label the eight basis vectors by the points of the projective line over the field with seven elements. First form the "Kirmse integers" : these consist of octonions whose coordinates are integers or half integers, and that are half integers (that is, halves of odd integers) on one of the 16 sets

∅ (∞124) (∞235) (∞346) (∞450) (∞561) (∞602) (∞013) (∞0123456) (0356) (1460) (2501) (3612) (4023) (5134) (6245)

of the extended quadratic residue code of length 8 over the field of two elements, given by , (∞124) and its images under adding a constant modulo 7, and the complements of these eight sets. Then switch infinity and any one other coordinate; this operation creates a bijection of the Kirmse integers onto a different set, which is a maximal order. There are seven ways to do this, giving seven maximal orders, which are all equivalent under cyclic permutations of the seven coordinates 0123456. (Kirmse incorrectly claimed that the Kirmse integers also form a maximal order, so he thought there were eight maximal orders rather than seven, but as Coxeter (1946) pointed out they are not closed under multiplication; this mistake occurs in several published papers.)

The Kirmse integers and the seven maximal orders are all isometric to the E8 lattice rescaled by a factor of 12. In particular there are 240 elements of minimum nonzero norm 1 in each of these orders, forming a Moufang loop of order 240.

The integral octonions have a "division with remainder" property: given integral octonions a and b ≠ 0, we can find q and r with a = qb + r, where the remainder r has norm less than that of b.

In the integral octonions, all left ideals and right ideals are 2-sided ideals, and the only 2-sided ideals are the principal ideals nO where n is a non-negative integer.

The integral octonions have a version of factorization into primes, though it is not straightforward to state because the octonions are not associative so the product of octonions depends on the order in which one does the products. The irreducible integral octonions are exactly those of prime norm, and every integral octonion can be written as a product of irreducible octonions. More precisely an integral octonion of norm mn can be written as a product of integral octonions of norms m and n.

The automorphism group of the integral octonions is the group G2(F2) of order 12,096, which has a simple subgroup of index 2 isomorphic to the unitary group 2A2(32). The isotopy group of the integral octonions is the perfect double cover of the group of rotations of the E8 lattice.