Orbits of the automorphism group of the exceptional Jordan algebra.

Abstract

Necessary and sufficient conditions for two elements of a reduced exceptional simple Jordan algebra a to be conjugate under the automorphism group Aut a of a are obtained. It was known previously that if :a is split, then such elements are exactly those with the same minimum polynomial and same generic minimum polynomial. Also, it was known that two primitive idempotents are conjugate under Aut a if and only if they have the same norm class. In the present paper the notion of norm class is extended and combined with the above conditions on the minimum and generic minimum polynomials to obtain the desired conditions for arbitrary elements of :a. In this paper we characterize the orbits of the automorphism group Aut a of a reduced exceptional simple Jordan algebra I = '(03, y) over a field 'D of characteristic not two or three. Since Jacobson [2, Theorem 9] has shown that in case the octonion (Cayley-Dickson) algebra D is split then such an orbit consists of exactly those elements with the same minimum polynomial and the same generic minimum polynomial, we shall restrict our attention to octonion division algebras. In particular, we shall assume 'I' is infinite. Recall that D(03, y) is the Jordan algebra of 3 x 3 symmetric matrices with entries in D with respect to the involution x I> y 1cty, y = diag {Yl, Y2, Y3}, 0 = Yi e (D, and with multiplication x y= I(xy +yx), xy the usual matrix multiplication. An element x E D(03, y) iS of the form (1) x = aieii aik] with aoi E , ai E , i = 1, 2, 3, where (i, j, k) is a cyclic permutation of (1, 2, 3) and a[ij] = yjaeij + yiceji. We have the mapping x i-x# defined by (2) x# = 2 (aak yykn(a?))eii + 2 (y1(ajak) -aia1)[jk] where x is as in (1) and n is the norm on D. If x#& 0 and x# = 0, we say x is of rank one. Also, we write xxy=(x+y)#-x#-y#. For x, ye 3 write T(x,y)=T(x.y) where T(z) is the usual trace. Received by the editors January 27, 1970. AMS 1969 Subject Classifications. Primary 1740.