KW-sections for Vinbergʼs θ-groups of exceptional type

Let k be an algebraically closed field of characteristic not equal to 2 or 3, let G be a simple algebraic group of type F4, G2 or D4 and let θ be a semisimple automorphism of G of finite order. In this paper we consider the θ-group (in the sense of Vinberg) associated to these choices; we classify the positive rank automorphisms via Kac diagrams and we describe the little Weyl group in each case. As a result we show that all θ-groups in types G2, F4 and D4 have KW-sections, confirming a conjecture of Popov in these cases.