  In this note, we unify and extend various concepts in the area of
$G$-complete reducibility, where $G$ is a reductive algebraic group. By results
of Serre and Bate--Martin--R\"{o}hrle, the usual notion of $G$-complete
reducibility can be re-framed as a property of an action of a group on the
spherical building of the identity component of $G$. We show that other
variations of this notion, such as relative complete reducibility and
$\sigma$-complete reducibility, can also be viewed as special cases of this
building-theoretic definition, and hence a number of results from these areas
are special cases of more general properties.
