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Exceptional Magic

The same thing, three waysTriality

In eight dimensions there are three different things with eight components: vectors, and two kinds of spinor. Triality is the symmetry that shuffles all three and leaves the arithmetic alone. It exists in eight dimensions and nowhere else.

First, the words.

A vector is an arrow: turn the world all the way round and an arrow comes back to itself. A spinor is the other kind of thing — turn the world all the way round and a spinor comes back with a minus sign; it takes two full turns to get home. Electrons are spinors. In eight dimensions, spinors come in two handednesses, and each one has eight components, same as the vector.

So you have three eight-dimensional objects that ought to be strangers, and they are not. There is one function of all three at once:

T(v, ψ, χ) = ( χ̃ , v ψ )
a real number out of three octonions: multiply two, conjugate the third, take the inner product. Lisi eq. (3.1).

and it does not care which of the three you call which:

T(v, ψ, χ) = T(ψ, χ, v) = T(χ, v, ψ)
checked on 4,000 random triples in exact integer arithmetic, worst disagreement 0

That cyclic symmetry is the octonion product, looked at from the side. You can go the other way and define the product from the form. Vector, spinor, other spinor: the labels come off.

Check it yourself

Three ways round

The last line reflects all three through a unit. A reflection turns the form into minus itself — the same size, the other sign — which is how you know a reflection is not one of the rotations. Two reflections make a rotation; four of them, through two different units, make a triality automorphism Lisi 2026 §3. Checked at build time on 1,920 exact cases and 3,000 random unit octonions, worst miss 2.1e-14.

A third of a turn

Twenty-four dots in three colours with three-fold symmetry: a third of a turn carries each colour onto the next.
The three eight-dimensional faces of so(8) — the algebra of rotations in eight dimensions — drawn in the plane where triality acts as a rotation. Twenty-four weights, three colours, 120 degrees apart.

Here is the same fact in pictures. The rotations of eight-dimensional space form an algebra called so(8), and it has three different eight-dimensional representations: 8v for the vectors, 8s and 8c for the two spinors. Triality is a symmetry of so(8) that cycles them.

The matrix that does it is built here from the diagram symmetry and then checked orthogonal; order3; permutes roots; 8v to 8c; 8c to 8s; 8s to 8v: it is orthogonal, it cubes to the identity, it permutes all 24 roots among themselves, and it carries 8v → 8c → 8s → 8v.

A three-pointed star with a rotation arrow, beside the three exceptional diagrams drawn as lines of nodes with one node branching off.
D4 is the only simple Lie algebra whose diagram has a three-fold symmetry. Three outer nodes around one centre; turn it a third and nothing changed.

That picture on the left is the whole reason triality exists. A Lie algebra's diagram records how its building blocks lean on each other, and symmetries of the diagram are symmetries of the algebra. Most diagrams have a mirror at best. D4 — the diagram of so(8) — is a three-pointed star, and you can turn it by a third.

Four algebras, four sizes of the same shape

Four boxes: su(2), su(3), sp(3) and f4, each showing its triality algebra plus three copies of a division algebra.
Each division algebra has a triality algebra, tri(D), and adding three copies of D to it gives a Lie algebra — su(2), su(3), sp(3), and f4 with 52 dimensions.

Every division algebra carries its own triality symmetry, and every one of them fits the same template: take the symmetry algebra, add one copy of the algebra for the vector and one for each spinor, and what comes out is a Lie algebra J. Evans 2009 C. H. Barton 2003 Lisi 2026 §4.

algebradimtri(𝔻)dim tri + three copiestotalwhich is
the reals ℝ1nothing03 × 1 = 33su(2)
the complex ℂ2u(1) + u(1)23 × 2 = 68su(3)
the quaternions ℍ4three copies of su(2)93 × 4 = 1221sp(3)
the octonions 𝕆8so(8)283 × 8 = 2452f4

The bottom row is the point: so(8) has 28 dimensions, three octonions make 24, and 28 + 24 = 52 — which is f4, an exceptional Lie algebra, built out of nothing but the octonions and their own symmetry.

Why anyone in physics cares

A circle with three spokes 120 degrees apart, labelled I, II and III, beside the rotation matrix that carries one spoke to the next.
A triality automorphism rotates each pair of eigenvectors by 120 degrees. Three positions, labelled I, II and III.

Matter comes in three generations. The electron, the muon and the tau are the same particle three times over with different masses, and nobody knows why there are three N. Furey 2025. Triality is a symmetry of order three that acts on exactly the objects spinors live in. The temptation is obvious, and it is old.

Whether the temptation pays off is a separate page, with the objections on it.