Let \(G=G(K)\) be a simple algebraic group defined over an algebraically closed field \(K\) of characteristic \(p\geq 0\). A subgroup \(X\) of \(G\) is said to be \(G\)-completely reducible if, whenever it is contained in a parabolic subgroup of \(G\), it is contained in a Levi subgroup of that parabolic. A subgroup \(X\) of \(G\) is said to be \(G\)-irreducible if \(X\) is in no proper parabolic subgroup of \(G\); and \(G\)-reducible if it is in some proper parabolic of \(G\). In this paper, the author considers the case that \(G=F_4(K)\).

The author finds all conjugacy classes of closed, connected, semisimple \(G\)-reducible subgroups \(X\) of \(G\). Thus he also finds all non-\(G\)-completely reducible closed, connected, semisimple subgroups of \(G\). When \(X\) is closed, connected and simple of rank at least two, he finds all conjugacy classes of \(G\)-irreducible subgroups \(X\) of \(G\). Together with the work of Amende classifying irreducible subgroups of type \(A_1\) this gives a complete classification of the simple subgroups of \(G\).

The author also uses this classification to find all subgroups of \(G=F_4\) which are generated by short root elements of \(G\), by utilising and extending the results of Liebeck and Seitz.