Octonions - Definition

Definition

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

with real coefficients {xi}.

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 product of all the terms, again like quaternions. The product of each term can be given by multiplication of the coefficients and a multiplication table of the unit octonions, like this one:

× e0 e1 e2 e3 e4 e5 e6 e7
e0 e0 e1 e2 e3 e4 e5 e6 e7
e1 e1 −e0 e3 −e2 e5 −e4 −e7 e6
e2 e2 −e3 −e0 e1 e6 e7 −e4 −e5
e3 e3 e2 −e1 −e0 e7 −e6 e5 −e4
e4 e4 −e5 −e6 −e7 −e0 e1 e2 e3
e5 e5 e4 −e7 e6 −e1 −e0 −e3 e2
e6 e6 e7 e4 −e5 −e2 e3 −e0 −e1
e7 e7 −e6 e5 e4 −e3 −e2 e1 −e0

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

The table can be summarized by the relations:

where is a completely antisymmetric tensor with value +1 when ijk = 123, 145, 176, 246, 257, 347, 365, and:

with e0 the scalar element, and i, j, k = 1 ... 7.

The above definition though is not unique, but is only one of 480 possible definitions for octonion multiplication. The others can be obtained by permuting the non-scalar elements, so can be considered to have different bases. Alternately they can be obtained by fixing the product rule for a few terms, and deducing the rest from the other properties of the octonions. The 480 different algebras are isomorphic, so are in practice identical, and there is rarely a need to consider which particular multiplication rule is used.

Read more about this topic:  Octonions

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