Structure of Chevalley groups: the proof from the Book

We describe and compare different geometric proofs of the main structure theorems for Chevalley groups over commutative rings. To warm up we sketch the known geometric proofs, published by I. Z. Golubchik, N. A. Vavilov, A. V. Stepanov and E. B. Plotkin, such as the $A_2$ and $A_3$ proofs for classical groups, $A_5$ and $D_5$ proofs for $E_6$; $A_7$ and $D_6$ proofs for $E_7$, and $D_8$ proof for $E_8$. After that we expound in more details the $A_2$ proofs for exceptional groups of types $F_4$, $E_6$ and $E_7$, based on multiple commutation. This new proof, the Proof from the Book, gives better bounds than any previously known. Moreover, unlike all previously known proofs it does not use results for fields, factorisation modulo radical, or any specific information concerning structure constants and equations defining exceptional Chevalley groups.