Does the Cayley–Dickson Construction expose new properties "The symmetries of the real field disappear as the Cayley–Dickson construction is repeatedly applied: first losing order, then commutativity of multiplication, associativity of multiplication, and next alternativity." - < Since the construction can be carried on ad infinitum, and a property is lost with each step, does that mean that a new property can be discovered with each step (the one that was just lost) infinitely? Somewhat related question: What specific algebraic properties are broken at each Cayley-Dickson stage beyond octonions?

Even though one loses a property at each of the first four Cayley-Dickson constructions does not imply that one can discover another one with each step. However, if one assumes that there are only finitely many symmetries that can be lost, then one should run out these after at some number Cayley-Dickson expansions/constructions.

It is a priory not known which if any other property is lost.