  Linear systems on Lie groups are a natural generalization of linear system on
Euclidian spaces. For such systems, this paper studies controllability by
taking in consideration the eigenvalues of an associated derivation D. When the
state space is a solvable connected Lie group, controllability of the systems
is guaranteed if the reachable set of the neutral element is open and the
derivation D has only pure imaginary eigenvalues. For bounded systems on
nilpotent Lie groups such conditions are also necessary.
