Jordan decomposition: Lie algebra

This lecture explores the Jordan decomposition in the context of Lie algebras, focusing on the concepts of semi-simple and unipotent elements. The instructor explains how every semi-simple element in the Lie algebra of a linear algebraic group is contained in a torus, while unipotent elements are contained in a unipotent subgroup. The lecture also covers the uniqueness of the Jordan decomposition, showing that every element in the Lie algebra of a linear algebraic group has a unique decomposition into semi-simple and unipotent parts. The proof involves constructing subgroups and analyzing commutativity to demonstrate the containment of the decomposition components in the Lie algebra of the group.