Consider a reductive group $G$ over a field $k$. The adjoint group $G^{\textrm{ad}}$ is defined by the exact sequence $$1\rightarrow Z(G)\rightarrow G\rightarrow G^{\textrm{ad}}\rightarrow 1$$The sequence $$1\rightarrow Z(G)(R)\rightarrow G(R)\rightarrow G^{\textrm{ad}}(R)\rightarrow 1$$ fails to be exact for a general $k$-algebra $R$, but is exact for $R=K$, the algebraic closure of $k$. Is there a criterion to determine for which $R$ the above sequence is exact? I'm mainly interested in the case that $k$ is a non-Archimedean local field, $G$ is a split $k$-group (namely $G=\textrm{SO}_{2n}$, $G^{\textrm{ad}}=\textrm{PSO}_{2n}$), and $R=k$.