## 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

### Summary

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.

### Mathematics Subject Classification

14L15, 20G10, 20G15

### Keywords/Phrases

linear algebraic groups, exceptional groups, torsors, non-Abelian cohomology, local regular rings, Grothendieck-Serre conjecture