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We present the complete classification of the tame irreducible elliptic quasi-simple Lie algebras of type Ai, and in particular, specialize on the case where the coordinates are not associative. Here the coordinates are Cayley-Dickson algebras over Laurent polynomial rings in v > 3 variables, which we call alternative tori. In giving our classification we need to present much information on these alternative tori and the Lie algebras coordinatized by them.doi:10.1090/s0002-9947-1995-1303115-1 fatcat:tchzentfkrflfcfa5obqmz3cee