The algebras to be
discussed are assumed to be finite dimensional and not necessarily associative.
If A is an algebra over a field K let Aut (A) denote the group of algebra
automorphisms of A. We define A to be doubly homogeneous if Aut (A) is
doubly transitive on the one-dimensional subspaces of A. Also a doubly
homogeneous algebra A is said to be nontrivial if A2≠0 and dimension A > 1. It is
shown that the only nontrivial doubly homogeneous algebra is unique up to
isomorphism.