Every Cayley-Dickson algebra can be viewed as a $(+,-,*,0,1)$ algebra. The reals and the complexes share the same equational identities. The quaternions have a subset of the equational identities, because they lose commutativity of multiplication. The octonions lose associativity, and the sedenions lose even more. Is the sedenions truly the last stop. Do all further Cayley-Dickson algebras have the same equational identities in that signature as the sedenions. This question is different from the tagged question, because I didn't see an answer for whether there are any equational identities that the sedenions have that any higher Cayley-Dickson algebras don't. I know that the sedenions are a power associative algebra.
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1The octonions lose general associativity, but they still have a restricted form of associativity: it's a so-called "alternative" algebra. The alternative property is like association, only two of the terms are equal, and one of those is the middle term: $$x(xy) = (xx)y\quad (xy)y = x(yy)$$The sedenions lose this property. Also, according to wiki, the construction may be carried out as many times as you wish, each time giving a new algebra that has twice the dimension of the last one. – Arthur Jun 30 '16 at 13:13