Lie Saturate and Controllability

We study the controllability of right-invariant bilinear systems on the complex and quaternionic special linear groups Sl(n,C) and Sl(n,H). The analysis relies on the Lie saturateLS(Γ), which characterizes controllability through convexity and closure properties of attainable sets, avoiding explicit Lie algebra computations. For Sl(n,C) with a strongly regular diagonal control matrix, we show that controllability is equivalent to the irreducibility of the drift matrix A, a property verified by the strong connectivity of its associated directed graph. For Sl(n,H), we derive controllability criteria based on quaternionic entries and the convexity of T2-orbits, which provide efficient sufficient conditions for general n and exact ones in the 2×2 case. These results link algebraic and geometric viewpoints within a unified framework and connect to recent graph-theoretic controllability analyses for bilinear systems on Lie groups. The proposed approach yields constructive and scalable controllability tests for complex and quaternionic systems.