Construction of groups with triality and their corresponding code loops

We generalize the global construction of code loops introduced by Nagy, which is based on the connection between Moufang loops and groups with triality. This follows from the construction of a nilpotent group $G_n$ of class 3 with triality and $2n$ generators, based on embeddings of $G_n$ into direct products of copies of $G_3$. In the finite case, where $G_n$ is a group such that $\mathrm{\mid }{G}_n\mathrm{\mid }=2^{4n+m}$ with $n\ge 3$ and $m=3\left(\genfrac{}{}{0px}{}{n}{2}\right)+2\left(\genfrac{}{}{0px}{}{n}{3}\right)$, we prove that the corresponding Moufang loop is the free loop $F_n$ with $n$ generators in the variety generated by code loops. The result depends on a construction similar to that of $G_n$, namely, embedding $F_n$ into direct products of copies of $F_3$, the free code loop associated with $G_3$.