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Is there a “conceptual” proof that an Albert algebra is a Jordan algbera?

That is, that $x(yx^2) = (xy)x^2$ ? In the original paper of A. Albert the proof is based on a direct calculation, so it is not even clear why $dim=3$ is important. References show that since then a ...
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A silly technical question on Albert algebras

Apologies in advance for this spectacularly uninteresting question, but it has just come up in my work. (Okay, not in a truly important way, but I am trying to gauge the scope of a certain ...