Part 6 of 9. The octonions are the last algebra in which lengths multiply. Their integral units are the 240 roots of E8, and pairing them with the other division algebras produces F4, E6, E7 and E8.
Part 5 found the E-type diagrams in groups of unit quaternions. This part doubles the quaternions once more, to the octonions.
The Cayley–Dickson construction makes each algebra from pairs of elements of the one before, with (a, b)(c, d) = (ac − d̄b, da + bc̄). Doubling ℝ gives ℂ, then the quaternions ℍ, then the eight-dimensional octonions 𝕆. In all four |ab| = |a| |b|, so division works, and Hurwitz proved in 1898 that no other algebra has such a norm.
An octonion has a real part and seven imaginary units e1, …, e7, each squaring to −1. Their products live on the Fano plane, seven points and seven lines with every pair of points on one line. Each line is a quaternion triple: along its arrow eiej = +ek, against it −ek.
Figure 1. The Fano plane in Baez's labelling: three sides, three medians and the inner circle are the seven lines, each arrow giving its triple's cyclic order. Click two points to multiply them, or a line below.
The plane decides which unit a product lands on and the arrows decide its sign, but not freely. Figure 2 reverses lines and tests whether the algebra stays alternative: (aa)b = a(ab) and (ab)b = a(bb), a law every normed algebra obeys.
| × | e₁ | e₂ | e₃ | e₄ | e₅ | e₆ | e₇ |
|---|
Figure 2. The products eiej for the line orientations listed on top; green is positive, red negative, shaded cells differ from Figure 1. Click a triple to reverse that line, or a square to load an orientation. Filled squares give an alternative algebra.
Of the 128 orientations, 16 give the octonions back: exactly those reached by replacing some units ep by −ep, which reverses the three lines through p. As 7-bit words they form the Hamming code of length 7; a parity bit extends it to the code H8 of Part 9. Over the 30 Fano planes on seven labelled units that makes the 480 tables usually quoted for octonion multiplication.
The associator $[a, b, c] = (ab)c - a(bc)$ measures the failure of associativity. Three units on one Fano line multiply inside a copy of ℍ and associate; three not on a line do not. So no algebra of matrices can hold the octonions faithfully.
Figure 3. The blue path computes $(ab)c$ and the red path $a(bc)$. When they end on the same node the triple associates.
Two elements are never enough: any two octonions lie in a copy of ℍ, so they generate an associative subalgebra. That gives alternativity, and the Moufang identities, such as $a(b(ac)) = ((ab)a)c$, extend the cancellation to three. Figure 4 tests five laws on each algebra the doubling builds, up to the 16-dimensional sedenions 𝕊.
Figure 4. Five laws tested on 600 random unit elements (seeded) of each algebra the doubling builds. Each cell is a histogram of |left side − right side|; "holds" means every sample agreed to rounding error. Click a cell to enlarge it.
ℝ to ℂ loses the ordering, which random tests cannot see. Each later doubling breaks one more law, and the step to 𝕊 breaks the rest at once, norm included: sedenions have zero divisors. For random unit octonions the mean of |[a, b, c]|² is exactly 21/16. The associator is trilinear, so the mean is the sum of |[ei, ej, ek]|² over the basis triples, 168 of them equal to 4, divided by 8³.
Since $|ab| = |a||b|$, the unit spheres $S^0, S^1, S^3, S^7$ carry a multiplication. They are also the only parallelizable spheres, with a smooth frame of tangent vectors at every point.
Figure 5. The four parallelizable spheres, one per normed division algebra. Sn−1 carries n − 1 independent tangent fields.
Inside $\mathbb{O}$ sits a maximal integral order, a ring of integer octonions, with exactly 240 units: the roots of $E_8$ scaled to length 1. The order is the $E_8$ lattice of Part 9. Figure 6 finds it by trying the 14 half-integer patterns of each of the 30 Fano planes until 240 units close under multiplication. They split by real part:
Figure 6. The 240 units of an integral octonion order, generated in the page and sorted by real part; click a band to list it. The status line reports the closure check and the inner products, only those of the E8 roots.
The automorphisms of 𝕆 form the 14-dimensional group G2, with Lie algebra Der(𝕆), the derivations. The other exceptional algebras pair two division algebras: Tits takes A and the Jordan algebra J = h3(B) of 3 × 3 Hermitian matrices over B, and builds
where primes mark the parts with zero real part or trace. dim A = 1, 2, 4, 8; dim Der(A) = 0, 0, 3, 14; dim J′ = 3 dim B + 2; and Der(h3(B)) is so(3), su(3), sp(3) or F4, of dimension 3, 8, 21, 52.
Figure 7. The Freudenthal–Tits magic square, computed from the dimension formula above; rows are A, columns B. Click a cell to see the sum. The formula treats A and B differently, yet the table is symmetric, and its bottom row is F4, E6, E7, E8.
With B = 𝕆, J is the 27-dimensional Albert algebra, whose derivations form F4. Part 7 follows it to F4, E6 and E7.