Totally Ramified Maximal Tori and Bruhat-Tits theory

Suppose $k$ is a nonarchimedean local field, $K$ is a maximally unramified extension of $k$, and $\mathbf{G}$ is a connected reductive $k$-group. If $\mathbf{T}$ is a $K$-minisotropic maximal $k$-torus in $\mathbf{G}$, then we use Bruhat-Tits theory to describe the stable classes in the $\mathbf{G}$-orbit of $\mathbf{T}$, the rational classes in the $\mathbf{G}$-orbit of $\mathbf{T}$, and the $k$-embeddings, up to rational conjugacy, into $\mathbf{G}$ of $\mathbf{T}$. We also provide, via Bruhat-Tits theory, a complete and explicit description of: the rational conjugacy classes of $K$-minisotropic maximal tame $k$-tori in $\mathbf{G}$; the stable classes of $K$-minisotropic maximal tame $k$-tori in $\mathbf{G}$; and the $k$-embeddings, up to rational conjugacy, into $\mathbf{G}$ of a $K$-minisotropic maximal tame $k$-torus of $\mathbf{G}$.