Adjoint quotients of reductive groups

Let $\text{rG}$ be a reductive group over a commutative ring $k$. In this article, we prove that the adjoint quotient $\text{adqG}$ is stable under base change. Moreover, if $\text{rG}$ has a maximal torus $\text{rT}$, then the adjoint quotient of the torus $\text{rT}$ by its Weyl group will be isomorphic to $\text{adqG}$. Then we focus on the semisimple simply connected group $\text{rG}$ of the constant type. In this case, $\text{adqG}$ is isomorphic to the Weil restriction $\underset{\rD/\spec k}{\prod}\aff^{1}_\rD$,where$\rD$istheDynkinschemeof$\rG$. Then we prove that for such $\text{rG}$, the Steinberg’s cross-section can be defined over $k$ if $\text{rG}$ is quasi-split and without ${\text{rA}}_{2m}$-type components