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. F4 is its symmetry algebra, E6 preserves its determinant, and E7 acts on a 56-dimensional space built from two copies of it. A second road to E6 and E7 runs through the 27 lines on a cubic surface.
F4 is the one Dynkin diagram with its double edge in the middle: four nodes in a chain, α1 and α2 long, α3 and α4 short. Its Cartan matrix is
Bn and Cn put the double edge at the end of the chain, leaving room for a family. In the middle, positivity allows two nodes on each side; a fifth at the long end gives the affine F̃4 of Part 4.
F4 has 48 roots in ℝ4. The 24 long roots have squared length 2 and the 24 short roots squared length 1:
Each set is the vertex set of a 24-cell, a regular 4-polytope whose 24 cells are octahedra, and the two are dual: the short roots point at the centres of the cells of the long 24-cell. The Weyl group W(F4), of order 1152, is the symmetry group of the 24-cell; it moves any root to any other of the same length, never to one of the other length.
Figure 1. The 24 long roots of F4 with the edges of the 24-cell, projected after a rotation in the x3x4 plane. Vertices that seem to pass through each other are moving past one another in the fourth dimension. The short roots give the dual 24-cell.
An element of h3(𝕆) has three real numbers on the diagonal and three octonions x, y, z off it, 3 + 24 = 27 real dimensions. The Jordan product X ∘ Y = ½(XY + YX) keeps them Hermitian. It is commutative and not associative; what survives is the Jordan identity (X ∘ Y) ∘ X2 = X ∘ (Y ∘ X2). Jordan, von Neumann and Wigner (1934) found four infinite families of such algebras built from associative ones, and this one exception.
The determinant survives as well:
where Re(xyz) is the same for either bracketing. Figure 2 tests the formula the hard way: every X satisfies its own cubic, X3 − (tr X) X2 + S(X) X − (det X) 1 = 0, with S the sum of the 2 × 2 principal minors. Read the triple product as Re(zyx) instead and the cubic fails for 28 of 40 seeded random elements.
Figure 2. An element X of h3(𝕆) with integer entries, editable (write octonions as 2 − e3 + e7), and a fixed second element Y. Each strip shows the 27 coordinates of one expression, computed exactly. The cubic and the Jordan identity come out zero; associativity does not.
The derivations of h3(𝕆) form F4:
the analogue of G2 = Der(𝕆), 52 dimensions where G2 has 14, and the start of the bottom row of the magic square.
Derivations preserve the trace as well as the determinant. The symmetries of det alone form E6, the reduced structure algebra: F4's 52 dimensions plus the 26 of the trace-zero matrices acting by Jordan multiplication, 78 in all. Its diagram is the (3, 3, 2) solution of Part 4's inequality, with 72 roots and a Weyl group of order 51,840.
Figure 3. The E6 diagram in Bourbaki labelling: the chain α1, α3, α4, α5, α6, with α2 attached to α4.
The elements with X ∘ X = X and trace 1 are the points of the 16-dimensional octonionic projective plane 𝕆P2, whose isometries form F4 and whose collineations form the real form E6(−26). The second button in Figure 2 loads three times such a point: X ∘ X − (tr X) X vanishes and det X = 0.
A smooth cubic surface in ℂP3 contains exactly 27 lines (Cayley and Salmon, 1849), and the symmetry group of their incidences is W(E6). Both facts come from one construction: the surface is the plane blown up at six points in general position. Each point becomes a curve ei, and every curve has a class D = a h − m1e1 − ⋯ − m6e6, with h a line, a the degree and mi the multiplicity at point i. Classes pair by
A line on the surface is a class with D · D = D · K = −1, and two lines meet when D · D′ = 1. Figure 4 finds every integer solution. Six points give 27: the six ei, the fifteen lines h − ei − ej through two points, and the six conics through five. The classes with D · D = −2 and D · K = 0 are the 72 roots of E6.
Figure 4. Every integer class D = a h − Σ miei solving the two equations on n blown-up points, by exhaustive search: one bar per degree a, with its pattern (a; m1 … mn). The line below checks each total against the Lie algebra.
Seven points give 56 exceptional curves and the 126 roots of E7; eight give 240 and 240 for E8. At nine, K · K = 0 and the search never closes, and the roots become E8's 240 repeated in every three consecutive degrees: the affine Ẽ8 that sat on the boundary of Part 4's inequality.
Figure 5. The 27 lines, joined wherever D · D′ = 1, from the classes of Figure 4: 135 meeting pairs. E1 starts at the top and the order runs clockwise through the Lij to C6. Select a line to see the 10 it meets.
The Albert algebra is the smallest representation of E6, for the real form E6(−26). The lines match it through its weights: the 27 weights form one W(E6) orbit, and joining two whose inner product is −2/3 gives the graph of Figure 5 exactly (a computer search finds an explicit matching). Under SO(10) the 27 splits as 16 + 10 + 1, with one generation of quarks and leptons in the 16, which is why E6 appears as a grand unified group (Gürsey, Ramond and Sikivie, 1976) after SU(5) and SO(10).
E7, the (4, 3, 2) diagram with dimension 133 and 126 roots, 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 det. E7 preserves it, and 56, the count of exceptional curves on seven points, is its smallest representation. Under E6, 56 = 1 + 1 + 27 + 27 and 133 = 78 + 1 + 27 + 27.
E8 has no representation smaller than its 248-dimensional adjoint, so Part 8 builds it from its 240 roots.