Let \(G\) be a simple exceptional algebraic group of adjoint type over the algebraic closure of a finite field. Let \(G_\sigma\) denote the fixed points of \(G\) where \(\sigma\) is some Frobenius endomorphism of \(G\). Let \(S\) denote the socle of \(G_\sigma\). One of the major problems in finite group theory is to describe the maximal subgroups of almost simple groups with socle \(S\). In this paper, the problem of describing the maximal subgroups \(M\) (up to conjugacy) if \(M\) is also a Chevalley group of the same characteristic and \(G\) has rank less than twice that of \(M\) is considered. The case where \(M\) is defined over a the field of \(q\) elements with \(q>2\) was settled by M. W. Liebeck, J. Saxl and D. M. Testerman [Proc. Lond. Math. Soc., III. Ser. 72, No. 2, 425-457 (1996; Zbl 0855.20040)].
In this paper, the authors complete the classification handling the case \(q=2\). This makes the theorem considerably more useful – since when applying the result, one would have to handle the cases not originally covered. The methods are considerably different in this case. Aside from a short list, the only examples are when \(M\) has maximal rank or is of the same type (possibly twisted) as \(G\). Indeed, the authors consider all embeddings of \(M\) into \(G\) (as was done in the earlier result), not just the maximal embeddings.