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The Cayley–Dickson octonions are alternative but not associative

Statement

For $O$ as in Real octonions by Cayley–Dickson doubling, $1_{O} = (1, 0)$ is a two-sided unit, both alternative laws $x (x y) = x^{2} y$ and $(y x) x = y x^{2}$ hold, and multiplication is not associative: $(i j) ℓ = k ℓ$ but $i (j ℓ) = - k ℓ$ . Thus $O$ is a unital alternative nonassociative real algebra.

Facts & Assumptions

Given: $x = (a, b)$ and $y = (c, d)$ in the pair model for $O$ .

[F1]

The Cayley–Dickson product is $(a, b) (c, d) = (a c - \overset{ˉ}{d} b, d a + b \overset{ˉ}{c})$ (Real octonions by Cayley–Dickson doubling).

[F2]

Quaternion multiplication is associative and $a \overset{ˉ}{a} = \overset{ˉ}{a} a = \hat{N (a)}$ , with $N (a) \in R$ ( $H$ is a division ring that is not commutative, hence not a field: $q^{- 1} = \overset{ˉ}{q} / N (q)$ for $q \neq 0$ , while $i j = k$ and $j i = - k$ ); embedded real scalars commute with quaternions (The quaternions $H$ : real quadruples with componentwise addition and an explicit multiplication formula matching the table on $1, i, j, k$ ).

[F3]

Octonion conjugation is an anti-involution, and for $a \in H$ , $a + \overset{ˉ}{a} = \hat{2 a_{0}}$ (Octonion conjugation is an anti-involution).

[F4]

An alternative algebra satisfies $x (x y) = x^{2} y$ and $(y x) x = y x^{2}$ (Alternative algebra).

[F5]

In quaternion coordinates the product has coordinates that are sums of signed terms $a_{p} b_{q}$ , and conjugation is $(a_{0}, a_{1}, a_{2}, a_{3}) \mapsto (a_{0}, - a_{1}, - a_{2}, - a_{3})$ (The quaternions $H$ : real quadruples with componentwise addition and an explicit multiplication formula matching the table on $1, i, j, k$ ).

Proof

1.1givenF1F2F3F5algebra

Each quaternion product coordinate is a sum of signed terms $a_{p} b_{q}$ , so the quaternion product is real-bilinear; the coordinate sign-change formula in [F5] makes conjugation real-linear. Therefore every term in the pair formula [F1] is bilinear in $x = (a, b)$ and $y = (c, d)$ , and the octonion product is real-bilinear. The quaternion coordinate formula also gives $\hat{1} a = a \hat{1} = a$ and $\overset{ˉ}{0} = 0$ , so [F1] gives $(1, 0) (a, b) = (a, b) = (a, b) (1, 0)$ . Let $t : = 2 a_{0} \in R$ . By [F2]–[F3], $a + \overset{ˉ}{a} = \hat{t}$ is central in $H$ and $b \overset{ˉ}{b} = \overset{ˉ}{b} b = \hat{N (b)}$ . Squaring $x$ in [F1] gives $x^{2} = (a^{2} - \hat{N (b)}, \hat{t} b)$ . Here the second coordinate is $b a + b \overset{ˉ}{a} = b (a + \overset{ˉ}{a}) = \hat{t} b$ .

1.2givenF1algebra

The quaternion table in The quaternions $H$ : real quadruples with componentwise addition and an explicit multiplication formula matching the table on $1, i, j, k$ gives $i j = k$ and $j i = - k$ . Formula [F1] gives $(i j) ℓ = (0, k) = k ℓ$ and $i (j ℓ) = (0, j i) = (0, - k) = - k ℓ$ . The pair $(0, k)$ is nonzero because its quaternion coordinate $k = (0, 0, 0, 1)$ is nonzero. Thus the two bracketings differ, so multiplication is not associative.

2.1givenF1F2step 1.1algebra

Substituting $x y = (a c - \overset{ˉ}{d} b, d a + b \overset{ˉ}{c})$ into [F1], expanding with associativity in [F2], and grouping $a + \overset{ˉ}{a} = \hat{t}$ gives $x (x y) = (a^{2} c - \hat{N (b)} c - \hat{t} \overset{ˉ}{d} b, d a^{2} - \hat{N (b)} d + \hat{t} b \overset{ˉ}{c}) .$ On the other hand, [F1] and step 1.1 give $x^{2} y = ((a^{2} - \hat{N (b)}) c - \overset{ˉ}{d} (\hat{t} b), d (a^{2} - \hat{N (b)}) + (\hat{t} b) \overset{ˉ}{c}),$ which is the same pair because real scalars are central. Thus $x (x y) = x^{2} y$ .

3.1givenF3step 2.1algebra

By [F3] and the left alternative law just proved, $\overline{(y x) x} = \overset{ˉ}{x} \overline{y x} = \overset{ˉ}{x} (\overset{ˉ}{x} \overset{ˉ}{y}) = ({\overset{ˉ}{x}}^{2}) \overset{ˉ}{y} = \overline{y (x^{2})} .$ The third equality uses the left alternative law on $\overset{ˉ}{x}, \overset{ˉ}{y}$ ; the fourth uses the anti-involution identities in [F3]. Since conjugation is an involution, it is injective, hence $(y x) x = y (x^{2})$ .

4.1step 1.1step 2.1step 3.1step 1.2F4∎

The product is real-bilinear by step 1.1, and steps 2.1 and 3.1 are exactly the two laws in [F4]; step 1.2 proves nonassociativity. Thus this is a unital alternative nonassociative real algebra.

Depends on

Used by

A concrete octonion associator witness Example

Cited to discharge well-definedness by Real octonions by Cayley–Dickson doubling.

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Sources

N. Jacobson, Lectures on Quadratic Jordan Algebras, §8 (standard reference, not scraped)
Holger P. Petersson, Albert Algebras, §3.5 (standard reference, not scraped)
John C. Baez, The Octonions, §2.2 (standard reference, not scraped)