Consider how the complex numbers are formed from the real numbers. If you generalise this carefully, then you can perform this operation again to get the quaternions, and so on. This operation is the Cayley–Dickson construction.
Let be an possibly nonassociative -algebra over the field of real numbers: an algebra equipped with an involution which is an antiautomorphism. (Actually, could be replaced by any commutative ring in the definitions, although some properties may depend on this ring.)
and the formula
defines an involutive antiautomorphism on , so the doubling procedure can be iterated.
The map is a monomorphism . If is unital with unit then is unital with unit . In the unital case, the element has the property , and we may write as (while ). For this reason, we may write in place of , at least when is unital.
Generally speaking, the double of an algebra has a nice property iff is one level nicer. For simplicity, assume that is unital (so that is a subalgebra). Since , we see that the involution on is trivial iff the involution on is trivial and further has . Since , is commutative iff is commutative and the involution in is trivial. Since , is associative iff is associative and commutative. Finally, is alternative iff is associative (and hence also alternative).
The standard example is the sequence of consecutive doubles starting with itself (with the identity map as involution); these are the Cayley–Dickson algebras: the real numbers , the complex numbers , the quaternions , the octonions or Cayley numbers , the sedenions , etc. These are the normed division algebras (, , , and ), followed by further algebras which are not division algebras. All of these algebras are power-associative and unital and have all inverse elements; the subalgebra with is always just .
M M Postnikov, Lectures on geometry, Semester V: Lie groups and Lie algebras, Lec. 14 (russian and english editions)
John Baez, This Week’s Finds — Week 59