Moufang Loops and Groups with Triality are Essentially the Same Thing

In 1925 Elie Cartan introduced the principal of triality specifically for the Lie groups of type D4, and in 1935 Ruth Moufang initiated the study of Moufang loops. The observation of the title was made by Stephen Doro in 1978 who was in turn motivated by work of George Glauberman from 1968. Here we make the statement precise in a categorical context. In fact the most obvious categories of Moufang loops and groups with triality are not equivalent, hence the need for the word “essentially.” Received by the editor 20 June 2016. 2010 Mathematics Subject Classification. Primary 20.