Split-octonion

 ( Composition Algebras ) 

Up to isomorphism, the octonions and the split-octonions are the only two 8-dimensional composition algebras over the real numbers. They are also the only two over the real numbers. Split-octonion algebras analogous to the split-octonions can be defined over any field.

$(a\; +\; b\; \backslash ell)(c\; +\; d\; \backslash ell)\; =\; (ac\; +\; \backslash lambda\; \backslash bar\; db)\; +\; (da\; +\; b\backslash bar\; c)\backslash ell$

$\backslash lambda\; =\; \backslash ell^2.$

Every split-octonion $x$ can be written as a linear combination of the basis elements,

$x\; =\; x\_0\; +\; x\_1\backslash ,i\; +\; x\_2\backslash ,j\; +\; x\_3\backslash ,k\; +\; x\_4\backslash ,\backslash ell\; +\; x\_5\backslash ,\backslash ell\; i\; +\; x\_6\backslash ,\backslash ell\; j\; +\; x\_7\backslash ,\backslash ell\; k,$

By linearity, multiplication of split-octonions is completely determined by the following multiplication table:

 !$i$

 !$j$

 !$k$

 !$\backslash ell$

 !$\backslash ell\; i$

 !$\backslash ell\; j$

 !$\backslash ell\; k$

A convenient mnemonic is given by the diagram at the right, which represents the multiplication table for the split-octonions. This one is derived from its parent octonion (one of 480 possible), which is defined by:

$e\_i\; e\_j\; =\; -\; \backslash delta\_\{ij\}e\_0\; +\; \backslash varepsilon\; \_\{ijk\}\; e\_k,\backslash ,$

where $\backslash delta\_\{ij\}$ is the Kronecker delta and $\backslash varepsilon\; \_\{ijk\}$ is the Levi-Civita symbol with value $+1$ when $ijk\; =\; 123,\; 154,\; 176,\; 264,\; 257,\; 374,\; 365,$ and:

$e\_ie\_0\; =\; e\_0e\_i\; =\; e\_i;\backslash ,\backslash ,\backslash ,\backslash ,e\_0e\_0\; =\; e\_0\; ,$

with $e\_0$ the scalar element, and $i,\; j,\; k\; =\; 1\; ...\; 7.$

The red arrows indicate possible direction reversals imposed by negating the lower right quadrant of the parent creating a split octonion with this multiplication table.

$\backslash bar\; x\; =\; x\_0\; -\; x\_1\backslash ,i\; -\; x\_2\backslash ,j\; -\; x\_3\backslash ,k\; -\; x\_4\backslash ,\backslash ell\; -\; x\_5\backslash ,\backslash ell\; i\; -\; x\_6\backslash ,\backslash ell\; j\; -\; x\_7\backslash ,\backslash ell\; k\; ,$

The quadratic form on x is given by

$N(x)\; =\; \backslash bar\; x\; x\; =\; (x\_0^2\; +\; x\_1^2\; +\; x\_2^2\; +\; x\_3^2)\; -\; (x\_4^2\; +\; x\_5^2\; +\; x\_6^2\; +\; x\_7^2)\; .$

This quadratic form N( x) is an isotropic quadratic form since there are non-zero split-octonions x with N( x) = 0. With N, the split-octonions form a pseudo-Euclidean space of eight dimensions over R, sometimes written R^4,4 to denote the signature of the quadratic form.

If N( x) ≠ 0, then x has a (two-sided) multiplicative inverse x⁻¹ given by

$x^\{-1\}\; =\; N(x)^\{-1\}\{\backslash bar\; x\}.$

$N(xy)\; =\; N(x)N(y).$

The automorphism group of the split-octonions is a 14-dimensional Lie group, the split real form of the exceptional simple Lie group G₂.

Define the "determinant" of a vector-matrix by the rule

.

$\backslash det(AB)\; =\; \backslash det(A)\backslash det(B).\backslash ,$

Zorn's vector-matrix algebra is, in fact, isomorphic to the algebra of split-octonions. Write an octonion $x$ in the form

$x\; =\; (a\; +\; \backslash mathbf\; v)\; +\; \backslash ell(b\; +\; \backslash mathbf\; w)$

• The Dirac equation in physics (the equation of motion of a free spin 1/2 particle, like e.g. an electron or a proton) can be expressed on native split-octonion arithmetic.M. Gogberashvili (2006) "Octonionic Electrodynamics", Journal of Physics A 39: 7099-7104.
• Supersymmetric quantum mechanics has an octonionic extension. V. Dzhunushaliev (2008) "Non-associativity, supersymmetry and hidden variables", Journal of Mathematical Physics 49: 042108 ;
• The Zorn-based split-octonion algebra can be used in modeling local gauge symmetric SU(3) quantum chromodynamics.B. Wolk, Adv. Appl. Clifford Algebras 27(4), 3225 (2017).
• The problem of a ball rolling without slipping on a ball of radius 3 times as large has the split real form of the exceptional group G₂ as its symmetry group, because this problem can be described using split-octonions.J. Baez and J. Huerta, G₂ and the rolling ball, Trans. Amer. Math. Soc. 366, 5257-5293 (2014); .

• R. Foot & G. C. Joshi (1990) "Nonstandard signature of spacetime, superstrings, and the split composition algebras", Letters in Mathematical Physics 19: 65–71
• F. Reese Harvey (1990). 9780123296504, Academic Press. ISBN 9780123296504
• Nash, Patrick L (1990) "On the structure of the split octonion algebra", Il Nuovo Cimento B 105(1): 31–41.
• T. A. Springer (2025). 9783540663379, Springer-Verlag. ISBN 9783540663379