An extension of a theorem of Nagano on transitive Lie algebras

Let M M be a real analytic manifold, and let L L be a transitive Lie algebra of real analytic vector fields on M M . A concept of completeness is introduced for such Lie algebras. Roughly speaking, L L is said to be complete if the integral trajectories of vector fields in L L are defined “as far as L L permits". Examples of situations where this assumption is satisfied: (i) L L = a transitive Lie algebra all of whose elements are complete vector fields, and (ii) L L = the set V ( M ) V(M) of all real analytic vector fields on M M . Our main result is: if M , M ′ M,M’ are connected manifolds, then every Lie algebra isomorphism F : L → L ′ F:L → L’ between complete transitive Lie algebras of real analytic vector fields on M , M ′ M,M’ which carries the isotropy subalgebra L m {L_m} of a point m m of M M to the isotropy s