Standard algebras
R. D. Schafer · Pacific Journal of Mathematics · 1969
In 1948 A. A. Albert defined a standard algebra Sί by the identities (x, y, z) + (z, x, y) -(x, z, y) = 0 and (x, y, wz) + O, y, xz) + (z, y, wx) = 0 .204 R. D. SCHAFER ( 4 ) (X, y, X 2 ) = 0.We shall define 21 to be a standard algebra in case (1), ( 2) and ( 4) are satisfied.Condition (4) is redundant except for characteristic 3. Put z = x in (1).Then( 5 ) (a?, y, x) = 0 for all x, y in 21; that is, SI is flexible.Hence, as Albert proved, every standard algebra is a noncommutative Jordan algebra [18, p. 140] and is therefore power-associative.The linearized form of ( 5) is (x, y, z) + (z, y, x) = 0 .Using flexibility, it is easy to see that, if an identity element 1 is adjoined to a standard algebra, the result is a standard algebra.Interchange x and z in (2), and subtract, in order to obtain ( 6 ) (w, y, [x, z\) = 0 18., An Introduction to nonassociative algebras, Academic Press, New York and London, 1966.