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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, there is a one-to-one correspondence between reduced crystallographic root systems and semi-simple Lie algebras. We show the construction of a root system from a semi-simple Lie algebra and conversely, the construction of a semi-simple Lie algebra from a reduced crystallographic root system. Let g be a semi-simple complex Lie algebra. Let further h be a Cartan subalgebra of g, i.e. a maximal abelian subalgebra. Then h acts on g via simultaneously diagonalizable linear maps in the adjoint representation. For in h define mathfrak{g}_lambda := {ainmathfrak{g}: [h,a]=lambda(h)atext{ for all }hinmathfrak{h}}.