Octonion

< 1 2 3 >

Numbers with eight dimensions are called octonions. You write them as

w = x₀ + x₁ i + x₂ j + x₃ k + x₄ l + x₅ il + x₆ jl + x₇ kl

and consists of the real numbers x_a and the units i, j, k, l, il, jl, kl .

Explanation

When multiplying the units i, j, k, l, il, jl, kl the order plays a role.

1 i j k l il jl kl

i −1 k −j il −l −kl jl

j −k −1 i jl kl −l −il

k j −i −1 kl −jl il −l

l −il −jl −kl −1 i j k

il l −kl jl −i −1 −k j

jl kl l −il −j k −1 −i

kl −jl il l −k −j i −1

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1	i	j	k	l	il	jl	kl
i	−1	k	−j	il	−l	−kl	jl
j	−k	−1	i	jl	kl	−l	−il
k	j	−i	−1	kl	−jl	il	−l
l	−il	−jl	−kl	−1	i	j	k
il	l	−kl	jl	−i	−1	−k	j
jl	kl	l	−il	−j	k	−1	−i
kl	−jl	il	l	−k	−j	i	−1