Sedenions

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

P(PQ) =(PP)Q

P^aP^b = P^a+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