(12) y = T(y)l - y and
(13) T(y) = Q(y,l) ,
defines a Jordan algebra structure of #, denoted ^(Q, 1) [19]. We say that
#(Q, 1) is the Jordan algebra of the quadratic form Q with base point 1.
We will be interested in finite dimensional simple Jordan algebra.
These have been determined in [16], (p. 206-210) if the characteristic is
not 2 and by McCrimmon if the characteristic is 2 (unpublished). We first
SECOND STRUCTURE THEOREM ([17], p. 3.59). Let £/$ be a simple
Jordan algebra satisfying the DCC for principal inner ideals. Then
# is one of the following types: (1) a Jordan division algebra. (2) an outer
ideal containing 1 in a Jordan algebra of a non-degenerate quadratic form with
base point over a field P/$, (3) an outer ideal containing 1 in W(G, *) where
(G, *) is simple associative with involution, (4) a reduced simple exception-
al Jordan algebra. Conversely, any algebra of one of the types (1) - (4)
satisfies the DCC for principal inner ideals and all of these are simple with
the exception of certain algebras of type (2) which are direct sums of two
division algebras isomorphic to outer ideals of P .
Algebras of type (4) are described explicitly in Chapter IV. Note that
if the characteristic is not 2, ideals and outer ideals coincide.
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