Construction of a family of Moufang loops

This paper is an excerpt from a Rayleigh essay submitted at the University of Cambridge in January 1970. We reproduce it now as it gives a general construction of a family of Moufang loops to which all bar one of the finite subloops of the Cayley algebra ${\mathbb O}$ belong. These subloops were classified up to isomorphism in the original essay, but are classified up to equivalence under the action of the group of symmetries of $\mathbb O$ in Boddington and Rumynin [1]. Explicitly, given a group G with an element a such that a 2=1 in its centre, we construct a Moufang group $\hat G$ in which G has index 2. $\hat G$ will be non-associative unless G is abelian.