Abstract
Different geometric proofs of the main structure theorems for Chevalley groups over commutative rings are described and compared. Known geometric proofs, published by I. Z. Golubchik, N. A. Vavilov, A. V. Stepanov, and E. B. Plotkin, such as A2 and A3 proofs for classical groups, A5 and D5 proofs for E6, A7 and D6 proofs for E7, and a D8 proof for E8 are given in outline. After that, A2 proofs for exceptional groups of types F4, E6, and E7, based on the multiple commutation, are discussed in more detail. This new proof, the proof from the Book, provides better bounds than any previously known proof. Moreover, it does not use results for fields, the factorization with respect to the radical, or any specific information concerning the structure constants and the equations defining exceptional Chevalley groups. Bibliography: 71 titles.
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