Isoparametric triple systems of FKM-type. I
[For the second part see the next review.] The largest and most important known class of isoparametric hypersurfaces in spheres are those considered in \textit{D. Ferus, H. Karcher, H. F. Münzner}, Math. Z. 177, 479-502 (1982; Zbl 0443.53037). The authors have developed an algebraic approach describing all isoparametric hypersurfaces in a sphere with 4 distinct principal curvatures by isoparametric triple systems [Tôhoku Math. J. 35, 187-224 (1983; Zbl 0507.53038) and ibid., 225-248 (1983; Zbl 0507.53039)]. In the present paper this general approach is specialized to those isoparametric triple systems which correspond to the surfaces constructed by D. Ferus, H. Karcher and H. F. Münzner and which are called of FKM-type. In particular, tripotents and their Peirce decompositions are described which are the most important tool in the general theory. Moreover, an intrinsic characterization for an isoparametric triple system to be of FKM-type is obtained. The results lay the foundations for the following 2 parts and are also used in the classification of isoparametric hypersurfaces of ''algebra type'', first studied by \textit{H. Ozeki} and \textit{M. Takeuchi} [ibid. 27, 515-559 (1975; Zbl 0359.53011) and ibid. 28, 7-55 (1976; Zbl 0359.53012)] and classified by the authors [Osaka J. Math. 20, 145-176 (1983)].
scientific article; zbMATH DE number 3842677