Starting from the variety of associative, Lie, Jordan, or alternative algebras with involution (or involutive automorphism), we classify all the formal bilinear products of the form axy+bx*y+cxy*+dx*y*+Ayx+Byx*+Cy*x+Dy*x*, formed with the help of the original product and the involution * (resp. involutive automorphism), which are either flexible, power-associative, alternative, associative, Jordan, binary-Lie, Malcev or Lie for all algebras of the chosen variety. To do so we formally define and study the change of product in an algebra acted on by a group of automorphisms and antiautomorphisms, solve the associative variety case via free algebras with computer assistance, and then solve the nonassociative cases via representations, nonassociative PI theory, and specific algebras.
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