On invariant theory of θ-groups

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Gradings of positive rank on simple Lie algebras (2012, methodology): “Using results of Panyushev in [23], we show thatCσ contains the automorphismθxm , wherexm is the point on the principal segment defined above.”
Vinberg’s θ-groups in positive characteristic and Kostant–Weierstrass slices (2009, methodology): “The solution in characteristic zero is then immediate due to Panyushev [ 14 ]; in positive characteristic we generalize Panyushev’s result by a similar reasoning.”
Torsion automorphisms of simple Lie algebras (2010, background): “Then the commutator g 7→ [g′, s′] induces a well-defined homomorphism As → π1(G) which is injective, since the centralizer of s′ in G′ is connected [21, 8.1].”
The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point (2012, background): “is a bijection, whose inverse is called the Kostant section [35].”