By Richard D. Schafer

Concise research offers in a brief area a number of the very important principles and leads to the idea of nonassociative algebras, with specific emphasis on replacement and (commutative) Jordan algebras. Written as an creation for graduate scholars and different mathematicians assembly the topic for the 1st time. "An vital addition to the mathematical literature"—Bulletin of the yankee Mathematical Society.

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**Example text**

Then (xy, xz) = (yx, zx), so that (22) ((xy)x, z) = (x(yx), z) for all x, y, z in S. Since (x, y) is nondegenerate on S, (22) implies (xy)x = x(yx); that is, S is flexible. To prove (c) we note first that (x, y) is a trace form on S+ : (23) (x · y, z) = (x, y · z) for all x, y, z in S. Also it follows from (23), just as in formula (14) of IV, that (24) (yS1 S2 · · · Sh , z) = (y, zSh · · · S2 S1 ) where Si are right multiplications of the commutative algebra S+ . In the commutative power-associative algebra S+ formula (4 ) becomes (25) 4x2 · (x · y) − 2x · [x · (x · y)] − x · (y · x2 ) − y · x3 = 0.

E. The algebra C3 of all 3 × 3 matrices with elements in a Cayley algebra C over F has the standard involution x → x (conjugate transpose). The 27-dimensional subspace H(C3 ) of self-adjoint elements (24) ξ1 c b c ξ2 a , b a ξ3 ξi in F , a, b, c in C, JORDAN ALGEBRAS 37 is a (central simple) Jordan algebra of degree t = 3 under the multiplication (23) where xy is the multiplication in C3 (which is not associative). Then J is any algebra such that some scalar extension JK ∼ = H(C3 )K (= H((CK )3 )).

Hence the minimal polynomial for Re divides f (λ), and the only possibilities for characteristic roots of Re are 1, 12 , 0 (1 must occur since e is a characteristic vector belonging to the characteristic root 1: eRe = e2 = e = 0). Also the minimal polynomial for Re has simple roots. Hence J is the vector space direct sum (10) where (11) J = J1 + J1/2 + J0 Ji = {xi | xi e = ixi } , i = 1, 1/2, 0. Taking a basis for J adapted to the Peirce decomposition (10), we see that Re has for its matrix relative to this basis the diagonal matrix diag{1, 1, .