Semigroups and Controllability of Invariant Control Systems on $mathrm{Sl}left(n,mathbb{H}right)$

Let $mathrm{Sl}left( n,mathbb{H}right)$ be the Lie group of $ntimes n$ quaternionic matrices $g$ with $leftvert det grightvert =1$. We prove that a subsemigroup $S subset mathrm{Sl}left( n,mathbb{H}right)$ with nonempty interior is equal to $mathrm{Sl}left( n,mathbb{H}right)$ if $S$ contains a subgroup isomorphic to $mathrm{Sl}left( 2,mathbb{H}right)$. As application we give sufficient conditions on $A,Bin mathfrak{sl}left( n,mathbb{H}right)$ to ensuring that the invariant control system $dot{g}=Ag+uBg$ is controllable on $mathrm{Sl}left( n,mathbb{H}right)$. We prove also that these conditions are generic in the sense that we obtain an open and dense set of controllable pairs $left( A,Bright)inmathfrak{sl}left( n,mathbb{H}right)^{2}$.