An extension of a theorem of nagano on transitive lie algebras

Let M be a real analytic manifold, and let L be a transitive Lie algebra of real analytic vector fields on M.A concept of completeness is introduced for such Lie algebras. Roughly speaking, L is said to be complete if the integral trajectories of vector fields in L are defined "as faras L permits". Examples of situations where this assumption is satisfied: (I) L → a transitive Lie algebra all of whoseelements are complete vector fields, and (ii) L - the set (M) of all real analytic vector fields on M. Our mainresult is: If M, M' ate connected manifolds, then every Liealgebra isomorphism F: L → L' between complete transitive Lie algebras of real analytic vector fields on M, M'which carries the isotropy subalgebra Lm of a point m of M to the isotropysubalgebra Lm' of m' ∊ M' is induced by a (unique) real analytic diffeomorphism f: M → M such that f(m) → m', provided that one of the following two conditions is satisfied: (1) M and M are simply connected, or (2) the Lie algebras L and L' separate points. Nagano had proved this result for the case L - V(M), L'- V(M), M and M' compact.