  We consider a control scheme where a quantum system S is put in contact with
an auxiliary quantum system A and the control can affect A only, while S is the
system of interest. The system S is then controlled indirectly through the
interaction with A. Complete controllability of S +A means that every unitary
state transformation for the system S +A can be achieved with this scheme.
Indirect controllability means that every unitary transformation on the system
S can be achieved. We prove in this paper, under appropriate conditions and
definitions, that these two notions are equivalent in finite dimension. We use
Lie algebraic methods to prove this result.
