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KW-sections for exceptional type Vinberg's $θ$-groups

Let $k$ be an algebraically closed field of characteristic not equal to 2 or 3, let $G$ be an almost simple algebraic group of type $F_4$, $G_2$ or $D_4$ and let $θ$ be an automorphism of $G$ of finite order, coprime to the characteristic. In this paper we consider the $θ$-group (in the sense of Vinberg) associated to these choices; we classify the positive rank automorphisms and give their Kac diagrams and we describe the little Weyl group in each case. As a result we show that all such $θ$-groups have KW-sections, confirming a conjecture of Popov in these cases.