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Consider a Cayley-Dickson algebra $(X,+,∗,0,1)$, that is an algebra generated from the reals by the Cayley-Dickson construction. From complexes to quaternions, we lose commutativity of multiplication, from quaternions to octonions, we lose associativity of multiplication, and from octonions to sedenions we lose alternativity of multiplication. I conjecture that, in a sense, the sedenions are the final stop. More precisely, for any Cayley-Dickson algebra $X$ that is sedenion or beyond, is the equational theory in the signature $(+,∗,0,1)$ for $X$ the same as the equational theory of the sedenions? I asked this question on math stack exchange, but I didn't receive an answer.

Consider a Cayley-Dickson algebra $(X,+,∗,0,1)$, that is an algebra generated from the reals by the Cayley-Dickson construction. From complexes to quaternions, we lose commutativity of multiplication, from quaternions to octonions, we lose associativity of multiplication, and from octonions to sedenions we lose alternativity of multiplication. I conjecture that, in a sense, the sedenions are the final stop. More precisely, for any Cayley-Dickson algebra $X$ that is sedenion or beyond, is the equational theory in the signature $(+,∗,0,1)$ for $X$ the same as the equational theory of the sedenions? I asked this question on math stackexchange over a week ago, but I didn't receive an answer.