Loops embedded in generalized Cayley algebras of dimension \(2^r\), \(r\geq 2\) (Q1599781)

Every finite-dimensional algebra over a field can be defined by the multiplication table of its basis. Such table is expressed by a matrix, the so-called multiplication matrix. There exists a class of real algebras called Cayley algebras of dimension \(2^r\), \(r\geq 2\) (for example the algebra of quaternions and the algebra of octonions are such algebras). It is shown that: (i) the basis of every Cayley algebra of dimension \(2^r\), \(r\geq 2\), forms a non-Abelian invertible loop of order \(2^{r+1}\), called a Cayley loop which is flexible and power-associative; (ii) all properties of a Cayley algebra are determined by its Cayley loop, but not all properties of the loop are satisfied by the algebra; (iii) the idea of the multiplication matrix can be used to construct other special algebraic structures like the group of Dirac operations in quantum electrodynamics.