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 finite groups of unit quaternions. This part goes one doubling past the quaternions, to the octonions, and follows them to the exceptional Lie algebras.
The complex numbers are pairs of reals, with i2 = −1. The Cayley–Dickson construction repeats that step: an element of the doubled algebra is a pair (a, b) of elements of the old one, multiplied by (a, b)(c, d) = (ac − d̄b, da + bc̄). Doubling ℝ gives ℂ, doubling ℂ gives the quaternions ℍ, and doubling ℍ gives the octonions 𝕆, an 8-dimensional algebra. At every one of these stages the norm is multiplicative, |ab| = |a| |b|, so there are no zero divisors and every nonzero element can be divided by. Hurwitz proved in 1898 that ℝ, ℂ, ℍ and 𝕆 are the only algebras with a multiplicative norm of this kind.
An octonion has a real part (e0 = 1) and seven imaginary units e1, …, e7, so there are 49 products to pin down. The Fano plane, the smallest projective plane, with seven points and seven lines, holds all of them. Any two distinct imaginaries ei, ej lie on exactly one line, and each line is a triple of units spanning a copy of the quaternions. The arrow on a line gives its cyclic order: going with it, ei · ej = +ek; going against it, ej · ei = −ek. The picture leaves out only the squares, which are ei · ei = −1 as in the quaternions.
Figure 1. The Fano plane. Seven imaginary units sit at the seven points; seven lines (three sides, three altitudes, and one inner circle) carry the multiplication rules. Every line is a cyclic quaternion triple (ei, ej, ek) with ei · ej = ek, ej · ek = ei, and ek · ei = ej. Reversing any of those flips the sign.
| × | e₁ | e₂ | e₃ | e₄ | e₅ | e₆ | e₇ |
|---|
Figure 2. The octonion multiplication table for the imaginary units. The diagonal is −1, and transposing any other cell flips its sign. Positive products are green, negative ones red. Hovering a cell loads that pair into the explorer above.
Reals, complexes and quaternions all satisfy $(ab)c = a(bc)$. Octonions do not, which is why they have no faithful matrix representation: matrix multiplication is associative. The associator $[a, b, c] = (ab)c - a(bc)$ measures the failure.
Three units on one Fano line multiply inside a copy of the quaternions, so they associate. Three distinct units not on a common line do not. Pick a triple below and compare the two groupings.
Figure 3. The blue path computes $(ab)c$ and the red path $a(bc)$. When they end on the same node the triple associates.
Any two octonions lie in a copy of the quaternions, so the subalgebra they generate is associative. With two elements, parentheses move freely; non-associativity needs three directions not on one Fano line.
Hurwitz's theorem makes the octonions the last algebra with $|ab| = |a||b|$. Doubling once more gives the 16-dimensional sedenions, where nonzero a, b can have ab = 0 and division fails. Each doubling up to 𝕆 gives up one property, and the step to the sedenions gives up two.
| Algebra | Ordered | Commutative | Associative | Alternative | Normed |
|---|
Figure 4. Properties lost by doubling. Each doubling from $\mathbb{R}$ to $\mathbb{O}$ doubles the dimension and loses one algebraic property; the step to $\mathbb{S}$ loses two. The octonions are the last step with a multiplicative norm.
Octonions still obey weaker laws, the Moufang identities, which make the nonzero octonions a Moufang loop. In each one the same element $a$ wraps the product on both sides, and the non-associativity cancels:
1. $a(b(ac)) = ((ab)a)c$ (left Moufang)
2. $((ca)b)a = c(a(ba))$ (right Moufang)
3. $(ab)(ca) = (a(bc))a$ (middle Moufang)
4. $(ab)(ca) = a((bc)a)$ (middle Moufang, other grouping)
Figure 5. Moufang identity verifier. Pick a law and test it on random elements. For octonions it always holds. Switch to the 16-dimensional sedenions and it fails: the Moufang laws imply alternativity, and the sedenions are not alternative.
Since $|ab| = |a||b|$, a product of unit elements is a unit element, so the unit spheres $S^0, S^1, S^3, S^7$ carry a multiplication. These are also the only parallelizable spheres, the ones with a smooth frame of independent tangent vectors at every point.
Figure 6. The four parallelizable spheres, one per normed division algebra. Each carries as many independent tangent fields as its own dimension.
Inside $\mathbb{O}$ sits a maximal integral order (a ring of integer octonions closed under multiplication) with exactly 240 units. Those 240 units are the 240 roots of $E_8$, scaled to length 1, and the order itself is the $E_8$ lattice, the densest packing in 8 dimensions (Part 9). Figure 7 builds the order: it tries the 14 half-integer patterns attached to each of the 30 ways of drawing a Fano plane on the seven units, and keeps the first set of 240 that is closed under multiplication.
The 240 units split by their real part:
Figure 7. The 240 units of an integral octonion order, generated and checked in the page, then sorted by scalar part. Click a band to see its members. The status line reports the closure check (every product of two units is again a unit) and the inner products between units, which take only the values of the E8 root system.
The automorphisms of 𝕆 form the 14-dimensional exceptional group G2, and its Lie algebra is Der(𝕆), the derivations of the octonions. The other exceptional algebras come from pairing two division algebras. Tits' construction takes a division algebra A and the Jordan algebra J = h3(B) of 3 × 3 Hermitian matrices over a second one, B, and builds a Lie algebra
where A′ and J′ are the parts with zero real part and zero trace. Every ingredient has a known dimension: dim A = 1, 2, 4, 8; Der(A) has dimension 0, 0, 3, 14; J′ has dimension 3 dim B + 2; and Der(h3(B)) is so(3), su(3), sp(3) or F4, of dimensions 3, 8, 21, 52. Figure 8 adds them up for every pair.
Figure 8. The Freudenthal–Tits magic square, computed from the dimension formula above. Rows are A, columns are B. Click a cell to see the sum. The formula treats A and B differently, yet the table comes out symmetric, and the bottom row is F4, E6, E7, E8.
With B = 𝕆 the Jordan algebra is the 27-dimensional Albert algebra h3(𝕆), whose derivations form F4. Part 7 follows that algebra to F4, E6 and E7, and Part 8 turns to E8 through its 240 roots.