Finite-dimensional Lie subalgebras of algebras with continuous inversion
  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. 
Daniel Beltita and Karl-Hermann Neeb