Dynkin diagrams do not classify global Lie groups

Assume the Axiom of Choice (The Axiom of Choice); it is inherited from the classification theorem cited below.

A connected Dynkin diagram classifies a finite-dimensional complex simple Lie algebra, and finite disjoint unions of connected Dynkin diagrams classify finite-dimensional complex semisimple Lie algebras. Neither classifies a real form or a connected Lie group with that Lie algebra (Cartan-Killing classification of complex simple Lie algebras, Semisimple algebras and disjoint unions of diagrams). Global classification of connected Lie groups requires isogeny or lattice data: connected groups with a given semisimple Lie algebra are classified by discrete central subgroups of the corresponding simply connected group, and that covering and lattice information is not visible in the diagram. Real semisimple Lie algebras require real-form data, namely an involution or a Satake diagram, because the diagram of the complexification does not distinguish the real forms. Both points are the subject of the following false statements and their refutations on this page; the classification of real forms and of global groups belongs to later pages and is not claimed here.