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Quaternions remove the commutative property and octonions eliminate the associative property can we go any higher and eliminate more properties?

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Yes, sedenions are the next step after octonions by not being an alternative algebra.

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  • $\begingroup$ Maybe I am asking too much, but is there yet another, and another... Or is there a limit? $\endgroup$ – BAR Mar 20 '13 at 4:33
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    $\begingroup$ My understanding is that the Cayley Dickson construction can always be used to create another number system, but that sedenions are the end of the line when it comes to removing properties. $\endgroup$ – Daniel Geisler Mar 20 '13 at 7:03
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You can have a look at Clifford algebra. But octonions are not Clifford algebra, which is associative.

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We would lose that it is a normed division algebra, as if it would be, than the octonions would be associative. In fact it is not normed nor a division algebra, so you lose all nice properties.

As Daniel said those are called the sedenions and are constructed by applying the cayley dickson construction to the octonions. They are not alternative, not normed and so on.

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