On rational and real elements in a class of Lie groups

For a class of groups $G$ over a field $\mathbb{F}$, including certain Lie groups, Algebraic groups and finite groups, we develop a general method to determine rational and real elements, thereby unifying earlier group-specific results into a wider framework. As an application, we classify all real and rational elements in the semidirect product $\mathrm{S}\mathrm{L}(2,\mathbb{R})⋉{\mathrm{S}\mathrm{y}\mathrm{m}}^n({\mathbb{R}}^2)$. Furthermore, for affine groups of the form $\mathrm{G}\mathrm{L}(n,\mathbb{R})⋉{\mathbb{R}}^n$, we show that if $x\in \mathrm{G}\mathrm{L}(n,\mathbb{R})$ is rational, then $(x,v)$ is rational for every $v\in {\mathbb{R}}^n$.