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The concept of isotopy plays an extremely important role in the structure theory of simple quadratic Jordan algebras satisfying the minimum condition on principal inner ideals. We take Koecher's characterization of isotopy and use it as the basis of a definition of semi-isotopy. It is clear that semi-isotopies induce, in a natural way, automorphisms of the lattice of inner ideals. We concern ourselves with the converse problem; namely, if 71 is a semilinear bijection of a quadratic Jordan algebra such that 71 induces an automorphism of the lattice of inner ideals, is 71 necessarily a semiisotopy? We answer the above question in the affirmative for a large class of simple quadratic Jordan algebras satisfying the minimum condition on principal inner ideals (said class includes all such algebras of capacity at least three over fields of characteristic unequal to two). Moreover, we prove that the only such maps which induce the identity automorphism on the lattice are the scalar multiplications. 1. Preliminaries. In this paper, we continue our study of automorphisms of the lattice of inner ideals of simple quadratic Jordan algebras begun in our earlier paper [7]. In that paper, attention was focused on determining the automorphism group of the lattice of inner ideals and on determining under what conditions two algebras can have isomorphic lattices of inner ideals. In the present paper our attention is focused on determining which semilinear bijections of a simple quadratic Jordan algebra satisfying the minimum condition on principal inner ideals induce automorphisms of the lattice of inner ideals. We will freely make use of our earlier results and of McCrimmon's determination of the inner ideals for the algebras under consideration [9]. The concept of an isotope of a quadratic Jordan algebra plays an extremely important role in the structure theory (see [3] ). One defines an isotopy to be an isomorphism of a quadratic Jordan algebra onto an isotope. Received by the editors July 20, 1973. AMS (MOS) subject classifications (1970). Primary 17C20, 17C30; Secondary 16A68.

Semi-isotopies and the lattice of inner ideals of certain quadratic Jordan algebras