1. The C.J.A. are nonassociative algebras defined by the relations: i)ab=ba, and ii) (a 2b)a=a 2 (ba). They are subdivided into: i) thespecial C.J.A., which are characterized by the productab=1/2(a·b+b·a)=1/2{a, b} (we calla·b the associative product), and ii) theexceptional C.J.A.,i.e. the algebras which are not special. For an extensive bibliography on C.J.A. seeH. Braun andM. Kocher:Jordan-Algebren (Berlin, 1966).
2. A. A. Albert:Trans. A.M.S.,64, 552 (1948).
3. The N.C.J.A. are nonassociative algebras neither anticommutative nor commutative defined by the relations: i)(ab)a=a(ba), and ii) (a 2b)a=a 2 (ba). They were first defined byR. D. Schafer:Proc. A.M.S.,6, 472 (1955). See also:Braun andKoecher (1):Jordan-Algebren (Berlin, 1966).R. D. Schafer:An Introduction to Nonassociative Algebras (New York, 1966);Proc. A.M.S.,9, 110 (1958);Trans. A.M.S.,94, 310 (1960);L. A. Kokoris:Proc. A.M.S.,9, 164 (1958);Canad. Journ. Math.,12, 448 (1960);L. J. Paige:Port. Math.,16, 15 (1957);R. H. Oehmke:Trans. A.M.S.,87, 226 (1958);Proc. A.M.S.,12, 151 (1961);K. McCrimmon:Pacific Journ. Math.,15, 925 (1965);Proc. A.M.S.,17, 1455 (1966);Trans. A.M.S.,121, 187 (1966).
4. The simple N.C.J.A. of characteristic zero (we consider only algebras and fields of characteristic zero) have been classified bySchafer (1955) (3) according to: i) the simple C.J.A.; ii) the simple quasi-associative algebras; iii) the simple flexible algebras of degree 2.
5. A. A. Albert:Proc. N.A.S.,35, 317 (1949).