Dynkin Diagrams and How they work

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Dynkin diagram, root system is a big mystery of my undergraduate life. Here I present three examples tha place they work. 

Warning

Since my main interest is not algebraic geometry, maybe there is something wrong or unacceptable, please inform me if you have good reference. This proof and concepts are collected from Shafarevich. 

Another topic which should be taken into consideration is McKay correspondence (using the conclusion of this section), if you have good recommendation (elementary or not), please inform me.

Contents

1. Definitions
1.1 Integral Quadratic Forms
1.2 Abstract Root systems
1.3 Dynkin Diagrams 
Exercises: Relative Concepts
1.4 Basis determine root system 
1.5 More about Weyl groups
1.6 Classification of Dynkin diagram
1.7 Euclidean diagrams and Quadratic forms again
References
2. How they work
2.1 Complex semisimple Lie algebras
2.2 Hereditary algebras
2.3 Classification of Du Val Singularities