  We investigate the finite-dimensional Lie groups whose points are separated
by the continuous homomorphisms into groups of invertible elements of locally
convex algebras with continuous inversion that satisfy an appropriate
completeness condition. We find that these are precisely the linear Lie groups,
that is, the Lie groups which can be faithfully represented as matrix groups.
Our method relies on proving that certain finite-dimensional Lie subalgebras of
algebras with continuous inversion commute modulo the Jacobson radical.
