Referee's report

[special-linear]

• p. 2, the line before Lemma 2.1, “when J is a one or two element set”, the correct punctuation
  here should be “ one- or two-element set”;
• p. 2, the line after formula 2.1, “The coefficients \eta can be expressed in terms of the structure
  constants”, probably it is better to write “in terms of the structure constants of the corresponding
  Lie algebra” or something like that;
• p. 3, the proof of the second claim of Lemma 2.3, it is not entirely clear what an s-shape is;
• p. 3, the paragraph after formula (2.2), the subgroup defined here should be called the
  “standard elementary parabolic subgroup” rather than just “standard parabolic subgroup”;
• p. 4, the proof of lemma 3.3, “which, contradicts”, superfluous comma;
• p. 5, the proof of Corollary 3.6, “denote by $\sigma$ the symmetry of the Dynkin diagram”,
  notation sigma is used later to denote the automorphism of $E(\Phi, R)$ induced by this
  symmetry which is confusing;
• p. 6, beginning of the proof of Theorem 1, the sentence “.. it suffices to consider only the
  following possibilities: ”, is incomplete. Apparently it refers to Table 1 which is on the next page.
• p. 7, the proof of Theorem 1, missed dot at the end of commutator formula for B_l.
• p. 8 the proof of Lemma 4.3, it is better to refer to the atlas of representations in the same
  sentence where the term “weight diagram” is used for the first time (as it is not introduced
  properly up to this point);
• p. 12 probably there should be some remark or logical break after the end of the proof of
  Theorem 5 (since the subsequent text has more relationship with the proof of Theorem 2);
• p. 13 the proof of Lemma 5.4, “we have a decomposition”, the article should be “the”; also the
  formula below looks more like a statement about inclusion of an element to a set rather than a
  group decomposition;
• p. 13, “it remains to note that...”, the order of summation and intersection is not specified, so it
  would be helpful to add some parentheses to the right-hand side;
• p. 14, line before Lemma 5.6, “Let $\mathcal{O}_S$ be a Dedekind ring of arithmetic type... “ --
  since there is only one Dedekind ring defined by S, the correct article here should be “the”;
• p. 15, statement of Theorem 6, Why use notation rk(Ф) for the rank of Ф while it could be simplified to just $\ell$? Also, the symbol $\ell$ is already used in the first claim of the theorem
  without being properly introduced.
• p. 15, in the definition of the width of a group, one has to either assume the generating set to be
  symmetric or add “and their inverses” in the end.