A convenient mnemonic for remembering the products of unit octonions is given by the following diagram at the right. This diagram with seven points and seven lines (the circle through i, j, and k is considered a line) is called the Fano plane. The lines are oriented in this diagram. The seven points correspond to the seven standard basis elements of Im(O). 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

ab = c and ba = −c

together with cyclic permutations. These rules together with

• 1 is the multiplicative identity,
• e² = −1 for each point in the diagram

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

Conjugate, norm, and inverse

The conjugate of an octonion

$x\; =\; x\_0\; +\; x\_1\backslash ,i\; +\; x\_2\backslash ,j\; +\; x\_3\backslash ,k\; +\; x\_4\backslash ,l\; +\; x\_5\backslash ,il\; +\; x\_6\backslash ,jl\; +\; x\_7\backslash ,kl$

is given by

$x\; =\; x\_0\; -\; x\_1\backslash ,i\; -\; x\_2\backslash ,j\; -\; x\_3\backslash ,k\; -\; x\_4\backslash ,l\; -\; x\_5\backslash ,il\; -\; x\_6\backslash ,jl\; -\; x\_7\backslash ,kl.$

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

The real part of x is defined as ½(x + x^*) = x₀ and the imaginary part as ½(x - x^*). The set of all purely imaginary octonions span a 7 dimension subspace of O, denoted Im(O).

The norm of the octonion x is defined as

$\backslash |x\backslash |\; =\; \backslash sqrt\{x^\{*\}\; x\}$

The square root is well-defined here as x^*x = xx^* is always a nonnegative real number:

$\backslash |x\backslash |^2\; =\; x^\{*\}x\; =\; x\_0^2\; +\; x\_1^2\; +\; x\_2^2\; +\; x\_3^2\; +\; x\_4^2\; +\; x\_5^2\; +\; x\_6^2\; +\; x\_7^2$

This norm agrees with the standard 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 is given by

$x^\{-1\}\; =\; \backslash frac\{x^\{*\}\}\{\backslash |x\backslash |^2\}$

It satisfies xx⁻¹ = x⁻¹x = 1.

Properties

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Octonionic multiplication is neither commutative:

$ij\; =\; -ji\; \backslash neq\; ji\backslash ,$

nor associative:

$(ij)l\; =\; -i(jl)\; \backslash neq\; i(jl)$

They do satisfy a weaker form of associativity: they are alternative. This means that the subalgebra generated by any two elements is associative. Actually, one can show that the subalgebra generated by any two elements of O is isomorphic to R, C, or H, all of which are associative.

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

$\backslash |xy\backslash |\; =\; \backslash |x\backslash |\backslash |y\backslash |$

This implies that the octonions form a nonassociative normed division algebra. The higher-dimensional algebras defined by the Cayley-Dickson construction (e.g. the sedenions) all fail to satisfy this property. They all have zero divisors.

Wider number systems exist which have a multiplicative modulus (e.g. 16 dimensional conic sedenions from the hypernumbers program). Their modulus is defined differently from their norm, and they also contain zero divisors.

It turns out that the only normed division algebras over the reals are R, C, H, and O. These four algebras also form the only alternative, finite-dimensional division algebras over the reals (up to isomorphism).

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

Automorphisms

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

A(xy) = A(x)A(y).

The set of all automorphisms of O forms a group called G₂. The group G₂ is a simply connected, compact, real Lie group of dimension 14. This group is the smallest of the exceptional Lie groups.

See also: PSL(2,7) - the automorphism group of the Fano plane.

Quotes

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The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative. — John Baez

See also

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• Split-octonions
• Hypercomplex numbers
  • Quaternions
  • Sedenions
• Hypernumbers
• Triality
• Spin(8)
• Seven dimensional cross product

References

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• John Baez, The Octonions, Bull. Amer. Math. Soc. 39 (2002), 145-205. Online HTML version at http://math.ucr.edu/home/baez/octonions/.
• John Conway and Derek Smith, On Octonions and Quaternions, A K Peters, Natick, MA (2003). ISBN 1-56881-134-9.

Octonions

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