Lie Saturate and Controllability

We study the controllability of right-invariant bilinear systems on the complex and quaternionic special linear groups (Formula presented.) and (Formula presented.). The analysis relies on the Lie saturate (Formula presented.), which characterizes controllability through convexity and closure properties of attainable sets, avoiding explicit Lie algebra computations. For (Formula presented.) 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 (Formula presented.), we derive controllability criteria based on quaternionic entries and the convexity of (Formula presented.) -orbits, which provide efficient sufficient conditions for general n and exact ones in the (Formula presented.) 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.