Infinite homogeneous algebras are anticommutative
Dokovic, D. Z.Sweet, Lowell G.
Proceedings of the American Mathematical Society
Mathematical and Computational Sciences
A (non-associative) algebra A, over a field k, is called homogeneous if its automorphism group permutes transitively the one dimensional subspaces of A. Suppose A is a nontrivial finite dimensional homogeneous algebra over an infinite field. Then we prove that x(2) = 0 for all x in A, and so xy = yx for all x; y is an element of A.