Moufang loops and alternative algebras

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.