Extends the classical Cayley-Dickson doubling from algebras to loops equipped with an involution, giving a construction that produces a doubled loop-with-involution from a given one and mirrors the algebra doubling that yields quaternions and octonions. Studies which varieties and identities of loops with involution are preserved under the doubling, and for central-by-abelian loops with elementary abelian 2-group quotients gives conditions determining the automorphism group of an iterated double. A key result is a corrected proof that the Cayley-Dickson loop Q_n has automorphism group GL_3(F_2) times {plus-or-minus 1}^(n-3) for n > 3.
Develops a theory of loops (a nonassociative generalisation of groups) equipped with an involution, and defines a Cayley-Dickson doubling operation on such loops that mirrors the classical doubling p…