On subloops of the loop of invertible elements of the split Cayley–Dickson algebra over a field that contain a subloop of transvections

Abstract

Let K be a field, O(K) the split Cayley–Dickson (octonion) algebra over K, and G(K) the Moufang loop formed by all elements of O(K) having norm other than 0. A transvection is an element of G(K) having the form $1+a,$ where $a\cancel{=}0$ and $a^2=0.$ In the paper, subloops of G(K), containing a subloop which consists only of transvections, are studied.

Keywords:

• Alternative algebras
• Moufang loops

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 20N05
• Secondary: 20N02