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The Sedenions form a 16-dimensional algebra over the reals obtained by applying the Cayley-Dickson construction to the octonions.

Like octonions, multiplication of sedenions is neither commutative nor associative. Unlike octonions, it does not even have the property of being "alternative". Multiplication is alternative if:

P(PQ) =(PP)Q

It does however have the property of being "power associative", since:

PaPb = Pa+b

for natural numbers a and b.

The sedenions have multiplicative inverses, but they are not a division algebra. This is because they have "zero divisors", i.e. there exist non-zero sedenions P, Q such that:

PQ = QP = 0