Every simple Lie algebra over ℂ belongs to one of four infinite families or is one of five exceptions. The series proves this classification, then follows the exceptions to E₈ and its lattice.
The finite-dimensional simple Lie algebras over ℂ fall into four infinite families and five exceptions, G₂, F₄, E₆, E₇ and E₈. The first four articles prove it: a Lie algebra becomes a set of root vectors, integrality leaves only four possible angles between them, the vectors compress into a Dynkin diagram, and a positivity test on diagrams leaves exactly nine answers. The next three follow the exceptions through the McKay correspondence, the octonions and the Albert algebra. The last two work with E₈ itself: its 240 roots as a polytope and as projections, and the lattice they generate, which is the densest sphere packing in eight dimensions.