Moufang loops and alternative algebras

Abstract:

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