  The present article is concerned with division algebras that are structurally
close to alternative algebras, in the sense that they satisfy some identity or
other algebraic property that holds for all alternative division algebras.
  Motivated by Belousov's ideas on quasigroups, we explore a new approach to
the classification of division algebras. By a detailed study of the
representations of the Lie group of autotopies of real division algebras we
show that, if the group of autotopies has a sufficiently rich structure then
the algebra is isotopic to an alternative division algebra. On the other hand,
it is straightforward to check that required conditions hold for large classes
of real division algebras, including many defined by identites expressable in a
quasigroup.
  Some of the algebras that appear in our results are characterized by the
existence of a well-behaved inversion map. We give an irredundant
classification of these algebras in dimension 4, and partial results in the
8-dimensional case.
