1. Semisimple Lie algebras:
Engel's theorem, Lie's theorem, Cartan's criterion, Semisimplicity of Lie algebras,
Representations of sl(2, F), Orthogonality + Integrality + Rationality.

2. Root systems:
Simple roots, Weyl chambers, Weyl group, Dynkin diagrams, Classification theorem.

3. Isomorphism and conjugacy theorems:
Isomorphism theorem, Cartan subalgebras, Conjugacy theorems, Conjugacy of Borel
subalgebras.

4. Existence theorem:
Universal enveloping algebra, Poincare-Birkhoff-Witt theorem, Serre's theorem, Criterion for
semisimplicity.

5. Representation theory:
Weights, Weight strings and weight diagrams, Casimir element, Freudenthal's formula,
Characters, Harish-Chandra's theorem, Kostant's multiplicity formula, Weyl's formula,.