Sedenions

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The sedenions form a 16-dimensional algebra over the real numbers and are obtained by applying the Cayley-Dickson Construction on the octonions.

As in the octonions, the multiplication of sedenions is neither commutative nor associative. But unlike the octonions, the sedenions do not even have the property of being an alternative algebra. However, they have the property of being power-associative.

Sedenions have 1 as a neutral element and inverses for multiplication, but they are not a division algebra, since they have divisors of zero.

Every sedenion is a linear combination of the unitary sedenions 1, e1, e2, e3, e4, e5, e6, e7, e8, e9 , e10, e11, e 12, e13, e14 and e >15, which form the basis of the vector space of the sedenions. The multiplication table of these unit sedenions is as follows.

× 1 e1e2e3e4e5e6e7e8e9e10e11e12e13e14e15
1 1 e1e2e3e4e5e6e7e8e9e10e11e12e13e14e15
e1e1−1 e3−e2e5−e4−e7e6e9−e8−e11e10−e13e12e15−e14
e2e2−e3−1 e1e6e7−e4−e5e10e11−e8−e9−e14−e15e12e13
e3e3e2−e1−1 e7−e6e5−e4e11−e10e9−e8−e15e14−e13e12
e4e4−e5−e6−e7−1 e1e2e3e12e13e14e15−e8−e9−e10−e11
e5e5e4−e7e6−e1−1 −e3e2e13−e12e15−e14e9−e8e11−e10
e6e6e7e4−e5−e2e3−1 −e1e14−e15−e12e13e10−e11−e8e9
e7e7−e6e5e4−e3−e2e1−1 e15e14−e13−e12e11e10−e9−e8
e8e8−e9−e10−e11−e12−e13−e14−e15−1 e1e2e3e4e5e6e7
e9e9e8−e11e10−e13e12e15−e14−e1−1 −e3e2−e5e4e7−e6
e10e10e11e8−e9−e14−e15e12e13−e2e3−1 −e1−e6−e7e4e5
e11e11−e10e9e8−e15e14−e13e12−e3−e2e1−1 −e7e6−e5e4
e12e12e13e14e15e8−e9−e10−e11−e4e5e6e7−1 −e1−e2−e3
e13e13−e12e15−e14e9e8e11−e10−e5−e4e7−e6e1−1 e3−e2
e14e14−e15−e12e13e10−e11e8e9−e6−e7−e4e5e2−e3−1 e1
e15e15e14−e13−e12e11e10−e9e8−e7e6−e5−e4e3e2−e1−1

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