**Original manuscript:** 2013/08/26

The report explores questions of orbits and controllability in driftless bilinear systems. Motivated by the Lie algebra rank approach to controllability, several results pertaining to orbit size and structure are presented. In particular, the prominence of invariant subspaces to system matrices is examined and a necessary condition for controllability is given. The condition is shown to be sufficient if and only if the system is two-dimensional. Finally, a particular class of driftless systems is explored and a graph-theoretic criterion for determining controllability of the systems in that class is given. Such a criterion is easily implementable as an algorithm of quadratic complexity.

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