  For a symmetric Lie algebra $\mathfrak g=\mathfrak k\oplus\mathfrak p$ we
consider a class of bilinear or more general control-affine systems on
$\mathfrak p$ defined by a drift vector field $X$ and control vector fields
$\mathrm{ad}_{k_i}$ for $k_i\in\mathfrak k$ such that one has fast and full
control on the corresponding compact group $\mathbf K$. We show that under
quite general assumptions on $X$ such a control system is essentially
equivalent to a natural reduced system on a maximal Abelian subspace $\mathfrak
a\subseteq\mathfrak p$, and likewise to related differential inclusions defined
on $\mathfrak a$. We derive a number of general results for such systems and as
an application we prove a simulation result with respect to the preorder
induced by the Weyl group action.
