Sedenion

The sedenions form a 16-dimensional algebra over the reals obtained by applying the Cayley-Dickson construction to the octonions.

Like octonions, multiplication of sedenions is neither commutative nor associative. But in contrast to the octonions, the sedenions do not even have the property of being alternative. They do, however, have the property of being power-associative.

The sedenions have a multiplicative identity element 1 and multiplicative inverses, but they are not a division algebra. This is because they have zero divisors.

Every sedenion is a real linear combination of the unit sedenions 1, e₁, e₂, e₃, e₄, e₅, e₆, e₇, e₈, e₉, e₁₀, e₁₁, e₁₂, e₁₃, e₁₄ and e₁₅, which form a basis of the vector space of sedenions. The multiplication table of these unit sedenions looks as follows.

* 1 e₁ e₂ e₃ e₄ e₅ e₆ e₇ e₈ e₉ e₁₀ e₁₁ e₁₂ e₁₃ e₁₄ e₁₅

1 1 e₁ e₂ e₃ e₄ e₅ e₆ e₇ e₈ e₉ e₁₀ e₁₁ e₁₂ e₁₃ e₁₄ e₁₅

e₁ e₁ -1 e₃ -e₂ e₅ -e₄ -e₇ e₆ e₉ -e₈ -e₁₁ e₁₀ -e₁₃ e₁₂ e₁₅ -e₁₄

e₂ e₂ -e₃ -1 e₁ e₆ e₇ -e₄ -e₅ e₁₀ e₁₁ -e₈ -e₉ -e₁₄ -e₁₅ e₁₂ e₁₃

e₃ e₃ e₂ -e₁ -1 e₇ -e₆ e₅ -e₄ e₁₁ -e₁₀ e₉ -e₈ -e₁₅ e₁₄ -e₁₃ e₁₂

e₄ e₄ -e₅ -e₆ -e₇ -1 e₁ e₂ e₃ e₁₂ e₁₃ e₁₄ e₁₅ -e₈ -e₉ -e₁₀ -e₁₁

e₅ e₅ e₄ -e₇ e₆ -e₁ -1 -e₃ e₂ e₁₃ -e₁₂ e₁₅ -e₁₄ e₉ -e₈ e₁₁ -e₁₀

e₆ e₆ e₇ e₄ -e₅ -e₂ e₃ -1 -e₁ e₁₄ -e₁₅ -e₁₂ e₁₃ e₁₀ -e₁₁ -e₈ e₉

e₇ e₇ -e₆ e₅ e₄ -e₃ -e₂ e₁ -1 e₁₅ e₁₄ -e₁₃ -e₁₂ e₁₁ e₁₀ -e₉ -e₈

e₈ e₈ -e₉ -e₁₀ -e₁₁ -e₁₂ -e₁₃ -e₁₄ -e₁₅ -1 e₁ e₂ e₃ e₄ e₅ e₆ e₇

e₉ e₉ e₈ -e₁₁ e₁₀ -e₁₃ e₁₂ e₁₅ -e₁₄ -e₁ -1 -e₃ e₂ -e₅ e₄ e₇ -e₆

e₁₀ e₁₀ e₁₁ e₈ -e₉ -e₁₄ -e₁₅ e₁₂ e₁₃ -e₂ e₃ -1 -e₁ -e₆ -e₇ e₄ e₅

e₁₁ e₁₁ -e₁₀ e₉ e₈ -e₁₅ e₁₄ -e₁₃ e₁₂ -e₃ -e₂ e₁ -1 -e₇ e₆ -e₅ e₄

e₁₂ e₁₂ e₁₃ e₁₄ e₁₅ e₈ -e₉ -e₁₀ -e₁₁ -e₄ e₅ e₆ e₇ -1 -e₁ -e₂ -e₃

e₁₃ e₁₃ -e₁₂ e₁₅ -e₁₄ e₉ e₈ e₁₁ -e₁₀ -e₅ -e₄ e₇ -e₆ e₁ -1 e₃ -e₂

e₁₄ e₁₄ -e₁₅ -e₁₂ e₁₃ e₁₀ -e₁₁ e₈ e₉ -e₆ -e₇ -e₄ e₅ e₂ -e₃ -1 e₁

e₁₅ e₁₅ e₁₄ -e₁₃ -e₁₂ e₁₁ e₁₀ -e₉ e₈ -e₇ e₆ -e₅ -e₄ e₃ e₂ -e₁ -1

Sedenion

Further reading