  Suppose $k$ is a nonarchimedean local field, $K$ is a maximally unramified
extension of $k$, and $\mathbf{G}$ is a connected reductive $k$-group. In this
paper we provide parameterizations via Bruhat-Tits theory of: the rational
conjugacy classes of $k$-tori in $\mathbf{G}$ that split over $K$; the rational
and stable conjugacy classes of the $K$-split components of the centers of
unramified twisted Levi subgroups of $\mathbf{G}$; and the rational conjugacy
classes of unramified twisted generalized Levi subgroups of $\mathbf{G}$. We
also provide parameterizations of analogous objects for finite groups of Lie
type.
