Variations on a Theme of Groups Splitting by a Quadratic Extension and Grothendieck-Serre Conjecture for Group Schemes $F_4$ with Trivial $g_3$ Invariant

Abstract

We study structure properties of reductive group schemes defined over a local ring and splitting over its étale quadratic extension. As an application we prove Serre--Grothendieck conjecture on rationally trivial torsors over a local regular ring containing a field of characteristic 0 for group schemes of type $F_4$ with trivial $g_3$ invariant.