Part 12 of 15. The last two simply-laced branched solutions. (4, 3, 2) gives E₇ with 126 roots; (5, 3, 2) gives E₈ with 240. The fourth hypothetical solution (6, 3, 2) lands at 1/p + 1/q + 1/r = 1 exactly, not strictly greater, and the exceptional E-branch stops there. E₈ is the ceiling of this branch, while the classical families continue indefinitely.
The two remaining Diophantine solutions (4, 3, 2) and (5, 3, 2) give E7 and E8. E7 ⊂ E8 as a root subsystem, and both share a Jordan-theoretic construction parallel to E6's. We spend most of this explainer on E7; the next three handle E8 in geometric detail: 240 roots and the Coxeter plane, sphere packing and the Leech lattice, and E8 in physics.
The E-series stops where it does because of arithmetic. A hypothetical "E9" with arms (6, 3, 2) gives 1/6 + 1/3 + 1/2 = 1 exactly, the boundary between finite-dimensional positive-definite Cartan matrices and infinite-dimensional positive-semidefinite ones. Lose strict inequality and you fall into affine Kac-Moody algebras.
The (4, 3, 2) arm triple gives 1 + 3 + 2 + 1 = 7 nodes, so E7 is rank 7. In Bourbaki labelling the main chain is α1, α3, α4, α5, α6, α7, with α2 branching off α4.
The structural numbers are:
E7 contains E6 by deleting α7, and E8 contains E7 by deleting α8. The chain E6 ⊂ E7 ⊂ E8 shrinks from 240 → 126 → 72 roots as we remove simple roots.
Like E6, E7 goes through h3(𝕆). E7 is built from the Freudenthal triple system: pairs (A, B, α, β) with A, B ∈ h3(𝕆) and α, β ∈ ℝ, total 56 dimensions, with a quartic form analogous to h3(𝕆)'s cubic determinant.
E7 preserves the quartic form, with a 56-dimensional fundamental representation. Under the E6 subgroup, that representation decomposes as 27 ⊕ 27* ⊕ 1 ⊕ 1. The Lie algebra itself decomposes as 𝔢7 = 𝔢6 ⊕ ℝ ⊕ 27 ⊕ 27*, so dim E7 = 133 comes from 78 + 1 + 27 + 27 = 133.
The geometric analogue of the Cayley plane is the Freudenthal variety, a 27-dim projective subvariety of ℙ55 with automorphism group E7. It is not strictly a projective plane (those stop at 𝕆P2) but carries the structural traces.
The (5, 3, 2) triple gives 1 + 4 + 2 + 1 = 8 nodes, so E8 is rank 8.
E8 is the largest exceptional Lie algebra by dimension and Weyl group order. Its root lattice is the densest sphere packing in 8 dimensions (Viazovska 2016).
The next three explainers of this atlas pivot from algebra to geometry: Part 13 lays out E8's 240 roots as the Gosset polytope 421 and projects them onto the Coxeter-plane mandala; Part 14 proves E8 is the densest 8-dimensional sphere packing (Viazovska 2016) and climbs to the Leech lattice in 24 dimensions; Part 15 catalogues where E8 surfaces in physics and coding theory.
Figure 1. The E-series terminator. Slide the longest arm length p from 4 to 6, with (q, r) fixed at (3, 2). At p = 4 we get E7 (sum = 13/12 ≈ 1.083). At p = 5 we get E8 (sum = 31/30 ≈ 1.033). At p = 6 the sum is exactly 1: the Cartan matrix becomes positive-semidefinite, one of its eigenvalues drops to zero, and the associated "Lie algebra" is no longer finite-dimensional. Instead, (6, 3, 2) corresponds to the affine Kac-Moody algebra Ẽ8, the simplest infinite-dimensional extension of E8.
E8 is the top of the E-series and the largest exceptional simple Lie algebra. (The classical families An, Bn, Cn, Dn are unbounded in dimension.) Relaxing the Diophantine inequality opens three increasingly exotic regimes:
Explainer 10 already placed the five exceptionals on the Freudenthal magic square.
The E-series terminates because arithmetic terminates. 1/p + 1/q + 1/r > 1 with p ≥ q ≥ r ≥ 2 is a finite problem; with q = 3, r = 2 the largest p is 5. Beyond that the inequality fails. "E8 is the largest exceptional Lie algebra" is arithmetic masquerading as algebra.