  Centreless Lie tori have been used by E. Neher to construct all extended
affine Lie algebras (EALAs). In this article, we study notions of isotopes and
isotopy for centreless Lie tori, and we use these notions, along with Neher's
construction, to show that there is a 1-1 correspondence between centreless Lie
tori up to isotopy and families of EALAs up to isomorphism. Also, centreless
Lie tori can be coordinatized by unital algebras that are in general
nonassociative, and, for many types of centreless Lie tori, there are classical
notions of isotopes and isotopy for the coordinate algebras. We show for those
types that an isotope of the Lie torus is coordinatized by an isotope of its
coordinate algebra. In writing the article, we have not assumed prior knowledge
of the theories of EALAs, Lie tori or isotopy. In fact, we hope that this
article will help to introduce the reader to these theories and their
interconnections.
