Simple Moufang loops and alternative algebras

TL;DR

This paper proves that if a Moufang loop contains a simple subloop, it cannot be embedded into the invertible elements of any alternative algebra, highlighting a structural limitation of such loops.

Contribution

It establishes a new non-embedding result for Moufang loops with simple subloops into alternative algebras.

Findings

Moufang loops with simple subloops cannot be embedded into invertible elements of alternative algebras.

The result clarifies structural restrictions of Moufang loops in algebraic contexts.

Provides insight into the relationship between loop substructures and algebra representations.

Abstract

Let a Moufang loop Q contain a non-unitary subloop, which is a simple loop. Then Q is not embedded into a loop of invertible elements of any alternative algebra.