Classification of algebras arising from the Tits process

Abstract

In the preceding paper [lo] we generalized the second Tits construction of simple exceptional Jordan algebras [6] to a construction over an arbitrary ring of scalars, which we called the Tits process, and claimed that it plays for Jordan rings a role akin to that of the Cayley-Dickson process for alternative rings. In this paper we assume that the scalars form a field F and determine the semisimple Jordan algebras arising from the Tits process. We see that, except for predictable problems in low characteristics, all simple Jordan algebras of degree 3 are obtainable by starting from Fl and repeating the Tits process. Review Let @ be a unital commutative associative ring, $ a @-module. Recall that JV = (N, #, 1) is a cubic norm structure on f if (1) N: f -+ @ is cubic form, # : 2 + f is a quadratic map, 1 is an element of 2, (2) # is an udjoint for N: