Moufang loops and alternative algebras

Abstract

Let O \mathbf {O} be the algebra O \mathbf {O} of classical real octonions or the (split) algebra of octonions over the finite field G F ( p 2 ) , p > 2 GF(p^2),\ p>2 . Then the quotient loop O ∗ / Z ∗ \mathbf {O}^*/Z^* of the Moufang loop O ∗ \mathbf {O}^* of all invertible elements of the algebra O \mathbf {O} modulo its center Z ∗ Z^* is not embedded into a loop of invertible elements of any alternative algebra.