Abstract
By means of differential geometry arguments Yang showed that for every unital real division algebra of dimension > 1 the square map
is onto and, consequently,
contains a subalgebra isomorphic to
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
with left-unit e containing no two-dimensional subalgebras with the additional property
Acknowledgement
The authors are very grateful to the Referee for his valuable comments which led to an improvement of the manuscript.