Editing Cayley–Dickson construction (section)

=== Further algebras ===
The Cayley–Dickson construction can be carried on ''[[ad infinitum]]'', at each step producing a power-associative algebra whose dimension is double that of the algebra of the preceding step. These include the 64-dimensional sexagintaquatronions (or 64-nions), the 128-dimensional centumduodetrigintanions (or 128-nions), the 256-dimensional ducentiquinquagintasexions (or 256-nions), and ''ad infinitum''.<ref>{{cite journal | last=Cariow | first=Aleksandr | title=An unified approach for developing rationalized algorithms for hypercomplex number multiplication | journal=Przegląd Elektrotechniczny | publisher=Wydawnictwo SIGMA-NOT | volume=1 | issue=2 | date=2015 | issn=0033-2097 | doi=10.15199/48.2015.02.09 | pages=38–41}}</ref> All the algebras generated in this way over a field are ''quadratic'': that is, each element satisfies a quadratic equation with coefficients from the field.<ref name=Sch66/>{{rp|50}}

In 1954, [[Richard D. Schafer|R. D. Schafer]] proved that the algebras generated by the Cayley–Dickson process over a field {{mvar|F}} satisfy the [[flexible identity]]. He also proved that any [[derivation algebra]] of a Cayley–Dickson algebra is isomorphic to the derivation algebra of Cayley numbers, a 14-dimensional [[Lie algebra]] over {{mvar|F}}.<ref>Richard D. Schafer (1954) "On the algebras formed by the Cayley–Dickson process", [[American Journal of Mathematics]] 76: 435–46 {{doi|10.2307/2372583}}</ref>