1 Historical Background and Informal Introduction to Lie Theory
2 Differentiable Manifolds, Differentiable Functions, Vector Fields, Tangent Spaces
3 Tangent Spaces; Mappings and Coordinate Representation

4 Affine Connections

Parallelism; Geodesics

Covariant Derivative
5 Normal Coordinates

Exponential Mapping
6 Definition of Lie Groups

Left-invariant Vector Fields

Lie Algebras

Universal Enveloping Algebra
7 Left-invariant Affine Connections

The Exponential Mapping

Taylor's Formula in a Lie Group Formulation

The Group GL (n, R)
8 Further Analysis of the Universal Enveloping Algebra

Explicit Construction of a Lie Group (locally) from its Lie Algebra

Exponentials and Brackets
9 Lie Subgroups and Lie Subalgebras

Closer Subgroups
10 Lie Algebras of some Classical Groups

Closed Subgroups and Topological Lie Subgroups
11 Lie Transformation Groups

A Proof of Lie's Theorem
12 Homogeneous Spaces as Manifolds

The Adjoint Group and the Adjoint Representation
13 Examples

Homomorphisms and their Kernels and Ranges
14 Examples

Non-Euclidean Geometry

The Associated Lie Groups of Su (1, 1) and Interpretation of the Corresponding Coset Spaces
15 The Killing Form

Semisimple Lie Groups
16 Compact Semisimple Lie Groups

Weyl's Theorem proved using Riemannian Geometry
17 The Universal Covering Group
18 Semi-direct Products

The Automorphism Group as a Lie Group
19 Solvable Lie Algebras

The Levi Decomposition

Global Construction of a Lie Group with a given Lie Algebra
20 Differential 1-forms

The Tensor Algebra and the Exterior Algebra
21 Exterior Differentiation and Effect of Mappings

Cartan's Proof of Lie's Third Theorem
22 Integration of Forms

Haar Measure and Invariant Integration on Homogeneous Spaces
23 Maurer-Cartan Forms

The Haar Measure in Canonical Coordinates
24 Invariant Forms and Harmonic Forms

Hodge's Theorem
25 Real Forms

Compact Real Forms, Construction and Significance
26 The Classical Groups and the Classification of Simple Lie Algebras, Real and Complex