Rationally isotropic exceptional projective homogeneous varieties are locally isotropic

Abstract

Assume that $R$ is a regular local ring that contains an infinite field and whose field of fractions $K$ has charactertistic $\ne 2$. Let $X$ be an exceptional projective homogeneous scheme over $R$. We prove that in most cases the condition $X(K)\ne \varnothing$ implies $X(R)\ne \varnothing$.