Part 7 of 9. Three exceptional Lie algebras grow out of the 27-dimensional Albert algebra h₃(𝕆): F₄ is its derivation algebra, E₆ preserves its cubic form, and E₇ acts on a 56-dimensional extension of it.
Part 6 ended with the exceptional Jordan algebra h3(𝕆) of 3 × 3 Hermitian octonionic matrices, whose derivations form F4. This part follows that algebra through F4, E6 and E7, the three exceptions between G2 = Aut(𝕆) and E8. The first, F4, has rank 4 and dimension 52, and its 24 long roots are the vertices of the 24-cell, a regular 4-polytope with no analogue in any other dimension.
In the classification of Part 4, F4 is the one non-simply-laced diagram with its double edge in the middle: a chain of four nodes, the edge between nodes 2 and 3 doubled, the arrow pointing from node 2 (long) to node 3 (short). Its Cartan matrix is
with −1 for each single edge, A23 = −2 and A32 = −1 for the double edge, and 0 between nodes that are not joined. Long and short roots differ in squared length by a factor of 2.
F4 has 48 roots in ℝ4. The 24 long roots have squared length 2:
These 4·C(4, 2) = 24 vectors are the vertices of a 24-cell and form a single orbit of the Weyl group W(F4), of order 1152. The 24 short roots have squared length 1:
The 8 vectors ±ei and the 16 vectors ½(±e1 ± e2 ± e3 ± e4) form a second 24-cell, dual to the first. Apart from the simplices, the 24-cell is the only self-dual regular polytope in dimension 3 or more.
Figure 1. The 24 long roots of F4 projected from 4D into 2D. The 24 vertices are connected by 96 edges at the minimum nonzero vertex-vertex distance. The slider rotates the polytope in the x3x4 plane before projecting, so vertices that seem to pass through each other are moving past one another in the fourth dimension.
The Albert algebra h3(𝕆) from Part 6 has general element
| a | z | ȳ |
| z̄ | b | x |
| y | x̄ | c |
Figure 2. A general element of h3(𝕆). The diagonal entries a, b, c are real (3 dimensions) and the three off-diagonal pairs each carry one octonion (3 × 8 = 24), for 27 in all. With the product A ∘ B = ½(AB + BA) this is a commutative, non-associative Jordan algebra.
Hermitian matrices are closed under the Jordan product even over the octonions, where the rest of matrix algebra breaks down. Jordan, von Neumann and Wigner (1934) classified the finite-dimensional formally real Jordan algebras: four infinite families that come from associative algebras, and one exception, h3(𝕆), which does not. Its derivation algebra is F4:
This is the analogue of G2 = Der(𝕆), with 14 dimensions of derivations on an 8-dimensional algebra becoming 52 on a 27-dimensional one. Adding the 26-dimensional trace-zero part of h3(𝕆) gives 52 + 26 = 78 = dim E6.
W(F4) is the full symmetry group of the 24-cell and acts transitively on the long roots, so the long roots are one orbit, and in 4D that orbit is the 24-cell. The same 24 vertices have two descriptions. They are the D4 root system ±ei ± ej, all of length √2. Scaled to unit length, they can also be taken as the 8 vertices ±ei of a 16-cell together with the 16 vertices (±½, ±½, ±½, ±½) of a tesseract. That both give a regular polytope happens only in 4D.
The classical types Bn and Cn put the double edge at the end of the chain, which leaves room for a family indexed by rank. F4 puts it in the middle, and positivity allows only two nodes on each side; a fifth node gives the affine diagram F̃4, which you can build in Part 4's sandbox.
As Der(h3(𝕆)), F4 starts the bottom row of the magic square: F4, E6, E7, E8 for 𝔸 = ℝ, ℂ, ℍ, 𝕆 with 𝔹 = 𝕆. It also sits inside E6 as the fixed points of an involution, and both have Coxeter number 12. The next step up, from the derivations of h3(𝕆) to its full structure algebra, is E6.
The branched solutions of 1/p + 1/q + 1/r > 1 are the family (n, 2, 2), giving Dn+2, and three more: (3, 3, 2), (4, 3, 2) and (5, 3, 2), giving the simply-laced exceptions E6, E7 and E8.
The (3, 3, 2) diagram has arms of 3, 3 and 2 nodes counting the shared centre, so 6 nodes in all. In Bourbaki labelling the main chain is α1, α3, α4, α5, α6, with α2 branching off α4. The Cartan matrix has −1 on edges and 0 elsewhere:
All roots have the same length. E6 has dimension 78, 72 roots, Coxeter number 12 and a Weyl group of order 51,840.
The E6 Dynkin diagram in the Bourbaki labelling: main chain α1, α3, α4, α5, α6, with α2 branching off α4. Six simple roots, five edges, a branch node of valency 3.
Derivations in F4 preserve the trace. E6 is the larger reduced structure algebra of h3(𝕆), the Lie algebra preserving the cubic form
which is well defined despite non-associativity. Its 78 dimensions are the 52 of F4 plus the 26 of the trace-zero matrices. The projectivisation of h3(𝕆) contains the 16-real-dimensional octonionic projective plane:
Non-associativity rules out octonionic projective spaces beyond dimension 2. F4 is the isometry algebra of 𝕆P², with F4/Spin(9) ≅ 𝕆P² in the same way that SO(3)/SO(2) = S2. E6 keeps the projective structure and drops the metric: its real form E6(−26) is the collineation group of 𝕆P².
A smooth cubic surface in ℂP³ contains exactly 27 lines (Cayley and Salmon, 1849), and each meets 10 of the other 26. The surface is ℂP² blown up at six points, and the lines split into 6 + 15 + 6:
From the blow-up picture the intersections are:
The automorphism group of this intersection graph is W(E6), of order 51,840, which nineteenth-century geometers (Jordan among them) studied as the group of the 27 lines.
Figure 3. The 27 lines on a smooth cubic surface, drawn as a graph with 27 nodes and 135 edges: 6 E-lines at the top, 15 L-lines in the middle arc, 6 C-lines at the bottom. Click a node to highlight the 10 lines it meets.
Three 27s have now appeared: the Albert algebra, the lines on a cubic surface, and the smallest representation of E6. Two of the links are exact. The Albert algebra is that representation, for the real form E6(−26). And the 27 weights of the representation form a single W(E6) orbit whose inner products reproduce the incidences of the 27 lines. The lines and the Albert algebra are related only through E6.
E6 also appears in particle physics as a grand unified gauge group (Gürsey, Ramond and Sikivie, 1976), the largest of the three after SU(5) (Georgi and Glashow, 1974) and SO(10). Under SO(10) its 27 splits as 16 + 10 + 1: one generation of quarks and leptons in the 16, plus extra states the model has to explain away.
The (4, 3, 2) diagram has 7 nodes. E7 has dimension 133, 126 roots, Coxeter number 18 and a Weyl group of order 2,903,040. Deleting α7 from its diagram leaves E6, and deleting α8 from E8 leaves E7, so the roots nest as 72 ⊂ 126 ⊂ 240.
E7 also grows out of the Albert algebra. Freudenthal's triple system is the 56-dimensional space of quadruples (α, β, A, B) with α, β scalars and A, B ∈ h3(𝕆), carrying a quartic form built from the cubic determinant. E7 is the algebra preserving that quartic form, and 56 is its smallest representation. Under E6 the pieces are visible: 56 = 1 + 1 + 27 + 27*, and the Lie algebra itself splits as 𝔢7 = 𝔢6 ⊕ ℂ ⊕ 27 ⊕ 27*, so 133 = 78 + 1 + 27 + 27.
That leaves E8, whose smallest representation is its own 248-dimensional adjoint. Part 8 builds it directly from its 240 roots.