In this paper we make various remarks, mostly of a computational nature, concerning a symplectic manifold X on which a Lie group G acts as a transitive group of symplectic automorphisms. The study of such manifolds was initiated by Kostant [41 and Souriau [5] and was recently developed from a more general point of view by Chu [2]. The first part of this paper is devoted to reviewing the Kostant, Souriau, Chu results and deriving from them a generalization of the Cartan conjugacy theorem. In the second part of this paper we apply these results to Lie algebras admitting a generalized (k, p) decomposition. In this paper we make various remarks, mostly of a computational nature, concerning a symplectic manifold X on which a Lie group G acts as a transitive group of symplectic automorphisms. The study of such manifolds was initiated by Kostant [4] and Souriau [5] and was recently developed from a more general point of view by Chu [2]. For the convenience of the reader we will begin by summarizing the basic facts. 1. General facts. Let G be a Lie group and X = G/H a homogeneous space for G where H is a closed subgroup, and let ir: G G/H = X be the projection. If Q is an invariant form on X then it is clear that a = 7T*Q is a left invariant form on G which satisfies (i) tI a = 0 for all t E h where h is the Lie algebra of H; (ii) a is invariant under right multiplication by elements of H, and hence under Ad for elements of H. Conversely, it is clear that any left invariant form a on G satisfying (i) and (ii) arises from G/H. If Q2 is a symplectic form then it is clear that a left invariant vector field will satisfy tla = 0 if and only if t E h. Furthermore, since do = 0, the set of all vector fields satisfying tla = 0 forms an integrable subbundle of TG, and in particular, the left invariant ones form a subalgebra of the Lie algebra of G; let us call it ha. We have thus recovered h. Let Ha be the group generated by ha. Notice that for any t E ha we have Dto = Ida + d(QJ1) = 0 so that a is invariant under Ha. The only problem is that Ha need not be closed. Let Received by the editors May 23, 1974. AMS (MOS) subject classifications (1970). Primary 53C15, 53C30. (1)This research was partially supported by NSF Grant GP43613X. Copyriglht