Adjoint quotients of reductive groups

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