A note on the construction of simple groups over number fields (Q1175601)

An approach to the classification of algebraic groups over arbitrary fields was developed by \textit{J. Tits} [Proc. Symp. Pure Math. 9, 33--62 (1966; Zbl 0238.20052)] and \textit{I. Satake} [Classification theory of semisimple algebraic groups (1971; Zbl 0226.20037)]. Tits (loc. cit.) has obtained an explicit classification of simple algebraic groups over local and number fields in terms of admissible \((k)\)-indices. The author observes that this construction does not cover all cases, as for exceptional groups it is not clear for which fields an admissible \((k)\)- index exists. In this paper an explicit field condition depending on the \((k)\)-index is introduced; furthermore, for fields satisfying this condition the author uses a way similar to the one of \textit{C. Chevalley} [Tôkoku Math. J., II. Ser. 7, 14--66 (1955; Zbl 0066.01503)] and of \textit{R. Steinberg} [Pac. J. Math. 9, 875--891 (1959; Zbl 0092.02505)] for constructing exceptional algebraic groups with a given \((k)\)-index. Proofs are outlined; a complete exposition will appear elsewhere.