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This is an exposition in order to give an explicit way to understand (1) anon-topological proof for an existence of a base of an affine root system, (2)a Serre-type definition of an elliptic Lie algebra with rank =>2, and (3) theisotropic root multiplicities of those elliptic Lie algebras.

Exposition on affine and elliptic root systems and elliptic Lie algebras