Part 11 of 15. Two rank-adjacent exceptions built on the exceptional Jordan algebra h₃(𝕆). F₄ (rank 4, dim 52) is the derivation algebra of the 27-dimensional Albert algebra and the isometry algebra of the octonionic projective plane 𝕆P². E₆ (rank 6, dim 78) enlarges F₄ by preserving the Albert algebra's cubic form, and its Weyl group is the symmetry group of the 27 lines on a smooth cubic surface.
After G2 the next exception is F4: rank 4, dimension 52, sitting between the classical B/C families and the E-series. It is the unique simple Lie algebra whose Dynkin diagram has a double edge in the middle of the chain.
Two facts: its 24 long roots are exactly the vertices of the 24-cell, the only self-dual regular 4-polytope, unique to dimension 4. And F4 is the derivation algebra of the exceptional Jordan algebra h3(𝕆), the 27-dimensional algebra of 3 × 3 Hermitian octonionic matrices.
F4 sits on the non-simply-laced branch of Act II. Pruning allowed chains with a middle double edge under bounded-length constraints; the finite check gives one shape:
The Cartan matrix of F4 is
Diagonal 2, off-diagonals −1 except A23 = −2 and A32 = −1. The squared-length ratio is 2, so |αlong|/|αshort| = √2.
F4 has 48 roots in 4D, split 24 long and 24 short. The long roots (squared length 2):
4·C(4, 2) = 24 vectors, the vertices of the 24-cell, each at distance √2. They form a single Weyl orbit under W(F4), a reflection group of order 1152.
The short roots (squared length 1) are the other 24 roots:
8 roots ±ei and 16 roots ½(±e1 ± e2 ± e3 ± e4), also a 24-cell dual to the long-root one. The 24-cell is self-dual, a unique feature of 4D.
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. Drag the slider or hit Play to rotate the polytope in its 4D ambient space. The rotation mixes the x3 and x4 coordinates before the 3D-to-2D projection, which is why vertices appear to pass through each other: they are actually moving "past" in the fourth dimension, while the 2D rendering can only show their shadow.
Just as G2 = Der(𝕆), F4 is the derivation algebra of the exceptional Jordan algebra h3(𝕆), the 27-dimensional algebra of 3 × 3 Hermitian octonionic matrices from Part 10. A general element:
| a | z | ȳ |
| z̄ | b | x |
| y | x̄ | c |
Figure 2. A general element of the 27-dimensional exceptional Jordan algebra h3(𝕆). The three diagonal entries a, b, c ∈ ℝ account for 3 real degrees of freedom; the three off-diagonal pairs (x, x̄), (y, ȳ), (z, z̄) each contribute one octonion (8 real degrees of freedom), so 3 × 8 = 24 more. Total: 3 + 24 = 27. Under the Jordan product A ∘ B = ½(AB + BA), this space forms a commutative (but non-associative) algebra, a Jordan algebra, and its derivation algebra is F4.
For associative matrices, Hermitian × Jordan = Hermitian. The same works formally for octonionic matrices, but octonion non-associativity breaks the usual matrix machinery. The Jordan product survives, and h3(𝕆) is the only such Jordan algebra that fails to be "special" (associative-derived).
This is the exceptional Jordan algebra: not from any associative construction. Jordan, von Neumann, and Wigner (1934) classified finite-dimensional Jordan algebras over ℝ as four infinite families plus the exception h3(𝕆).
And its derivation algebra is F4:
The analogue of G2 = Der(𝕆): G2 is 14-dimensional (derivations of an 8-dim algebra), F4 is 52-dimensional (derivations of 27-dim). 52 + 26 = 78 = dim(E6), where the extra 26 dimensions are the trace-zero part of h3(𝕆).
Why are the 24 long roots exactly the 24-cell? W(F4) acts transitively on the long roots and is the full symmetry group of the 24-cell. So the long roots are a single Weyl orbit, and in 4D the only polytope this orbit traces is the 24-cell.
The 24-cell has 24 vertices that can be described cleanly two ways. One: they are the D4 root system — the 24 vectors ±ei ± ej in ℝ4 (all of length √2). Two: scale to unit length and decompose as the 8 vertices of a 16-cell (±ei) together with the 16 vertices of a unit tesseract ((±½, ±½, ±½, ±½), all sign combinations). That both models give a regular polytope is a coincidence specific to 4D with no analogue in other dimensions.
F4's root system and the 24-cell are the same data: 48 roots = vertices of two dual 24-cells.
Why F4 is exceptional. Classical types Bn, Cn have the double edge at the end of the chain, allowing a family indexed by rank. F4 has it in the middle, where the valency constraint forces at most two nodes on each side, giving rank 4 and no analogue.
F4 connects to the other exceptionals in two ways. As Der(h3(𝕆)) it builds the E-series via the magic square: F4, E6, E7, E8 for 𝔸 = ℝ, ℂ, ℍ, 𝕆 with 𝔹 = 𝕆.
And F4 embeds in E6 as the fixed points of an involution. Both have Coxeter number 12, and the Cayley plane is an F4-homogeneous space inside the larger E6 story.
Next we climb one step up, from the derivation algebra of h3(𝕆) to the full structure algebra, and meet E6.
The simply-laced exceptions come from the three branched solutions of 1/p + 1/q + 1/r > 1: (3, 3, 2), (4, 3, 2), (5, 3, 2) — E6, E7, E8. This section covers E6: its root system, its connection to the octonionic projective plane, and the fact that W(E6) is the symmetry group of the 27 lines on a cubic surface.
The (3, 3, 2) diagram has three arms of lengths 3, 3, 2 from a central branch. Total nodes: 1 + 2 + 2 + 1 = 6, so E6 has rank 6.
In Bourbaki labelling, the main chain is α1, α3, α4, α5, α6, with α2 branching off α4. The Cartan matrix is the standard ADE pattern, −1 on edges and 0 elsewhere:
E6 is simply-laced, so all roots have equal length and no off-diagonal is more negative than −1. Rank 6, dim 78, |Δ| = 72, Coxeter number 12, |W| = 51,840.
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.
E6 builds on h3(𝕆). F4 derivations preserve the trace; E6 is the full reduced structure algebra, the Lie algebra preserving the cubic form
det is a well-defined cubic on h3(𝕆) despite octonion non-associativity. The 78-dimensional E6 is F4 (52) plus 26 dimensions of scaling and reshaping.
The projectivisation of h3(𝕆) contains the 16-real-dimensional octonionic projective plane:
Projective spaces over 𝕆 only exist up to dimension 2 because of non-associativity. F4 is the isometry algebra of 𝕆P², with F4/Spin(9) ≅ 𝕆P² (16-real-dimensional) — the same homogeneous space construction as SO(3)/SO(2) = S2. E6 is the larger symmetry that preserves not just the metric but the full projective-cubic structure: it is the derivation algebra of h3(𝕆) extended by the 26-dimensional trace-zero part, acting on the full Albert algebra.
A smooth cubic surface in ℂP³ contains exactly 27 complex lines (Cayley and Salmon, 1849). Each line meets exactly 10 of the other 26 and misses 16.
The 27 lines' intersection graph is a 10-regular graph on 27 vertices. The cleanest description uses the six points blown up to construct the cubic. The 27 lines split into 6 + 15 + 6:
The intersection rules are a short combinatorial table, derivable from the blow-up picture:
Each line has exactly 10 intersecting partners. The Weyl group W(E6) of order 51,840 is the full automorphism group of this graph, sometimes called the group of the 27 lines. Cayley, Jordan, and Klein knew this in the late 19th century.
Figure 3. The 27 lines on a smooth cubic surface, drawn as a graph with 27 nodes and 135 edges (each line meets exactly 10 others). The three groups of nodes, 6 E-lines at the top, 15 L-lines in the middle arc, 6 C-lines at the bottom, correspond to the "blow up ℂP² at 6 points" construction of the cubic surface. Click any node to see its 10 adjacent lines (yellow edges + highlighted nodes). The Weyl group of E6, of order 51,840, is the full automorphism group of this incidence pattern.
E6 appears as a candidate grand unified theory (GUT) gauge group. The Standard Model has gauge group SU(3) × SU(2) × U(1). GUT candidates: SU(5) (Georgi–Glashow 1974), SO(10) (1975), E6 (Gürsey–Ramond–Sikivie 1976). E6 is the smallest simple Lie algebra whose adjoint contains the Standard Model with the right fermion content: one generation of quarks and leptons fits into the 27-dim fundamental representation.
27 quarks-and-leptons ↔ 27 dimensions of h3(𝕆) ↔ 27 lines on a cubic surface. The physical relevance is open, but the math is rigid.
Three 27s. dim h3(𝕆) = 27, lines on a cubic surface = 27, dim of E6's fundamental = 27. Same object from three angles: Jordan coordinates, cubic-surface lines, E6 weights.
Next we take the last step up the E-series: E7 with its Freudenthal triple system, and E8, the ceiling of the exceptional E-branch.