Algebraic extensions of some results by Yang

Abstract

By means of differential geometry arguments Yang showed that for every unital real division algebra A of dimension > 1 the square map x ↦ x 2 is onto and, consequently, A contains a subalgebra isomorphic to C . The first assertion is extended algebraically for an algebra with only a non-zero flexible element. It persists if the unit is replaced by a non-zero central element. Next, always by algebraic approach, we give examples of both four-dimensional and eight-dimensional real division algebras A with left-unit e containing no two-dimensional subalgebras with the additional property R e 2 = I A .