Abstract
In this chapter, we shall show that for each isomorphism class of irreducible root systems there is a unique simple Lie algebra over C (up to isomorphism) with that root system. Moreover, we shall prove that every simple Lie algebra has an irreducible root system, so every simple Lie algebra arises in this way. These results mean that the classification of irreducible root systems in Chapter 13 gives us a complete classification of all complex simple Lie algebras.
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