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Alternative twisted tensor products and Cayley algebras

We introduce what we call "alternative twisted tensor products" for not necessarily associative algebras, as a common generalization of several different constructions: the Cayley-Dickson process, the Clifford process and the twisted tensor product of two associative algebras, one of them being commutative. We show that some very basic facts concerning the Cayley-Dickson process (the equivalence between the two different formulations of it and the lifting of the involution) are particular cases of general results about alternative twisted tensor products of algebras. As a class of examples of alternative twisted tensor products, we introduce a "tripling process" for an algebra endowed with a strong involution, containing the Cayley-Dickson doubling as a subalgebra and sharing some of its basic properties.