Finite subloops of units in an alternative loop ring

An RA loop is a loop whose loop rings, in characteristic different from $2$, are alternative but not associative. In this paper, we show that every finite subloop $H$ of normalized units in the integral loop ring of an RA loop $L$ is isomorphic to a subloop of $L$. Moreover, we show that there exist units $\gamma_i$ in the rational loop algebra $\mbox{\sf\bf Q}L$ such that $\gamma_k^{-1}(\ldots(\gamma_2^{-1} (\gamma_1^{-1}H\gamma_1)\gamma_2)\ldots) \gamma_k\subseteq L$. Thus, a conjecture of Zassenhaus which is open for group rings holds for alternative loop rings (which are not associative).