a maximal compact subgroup can be found in and . and discuss the extension to connected Lie groups and connected locally compact groups. For semisimple groups, existence is a consequence of the existence of a compact real form of the noncompact semisimple Lie group and the corresponding Cartan decomposition. The proof of uniqueness relies on the fact that the corresponding Riemannian symmetric space G/K has negative curvature and Cartan's fixed point theorem. showed that the derivative of the exponential map at any point of G/K satisfies |d exp X| ≥ |X|. This implies that G/K is a Hadamard space, i.e. a complete metric space satisfying a weakened form of the parallelogram rule in a Euclidean space. Uniqueness can then be deduced from the Bruhat-Tits fixed point theorem. Indeed, any bounded closed set in a Hadamard space is contained in a unique smallest closed ball, the center of which is called its circumcenter. In particular a compact group acting by isometries must fix the circumcenter of each of its orbits. Proof of uniqueness for semisimple groups also related the general problem for semisimple groups to the case of GL(n, R). The corresponding symmetric space is the space of positive symmetric matrices. A direct proof of uniqueness relying on elementary properties of this space is given in . Let be a real semisimple Lie algebra with Cartan involution σ. Thus the fixed point subgroup of σ is the maximal compact subgroup K and there is an eigenspace decomposition where , the Lie algebra of K, is the +1 eigenspace. The Cartan decomposition gives If B is the Killing form on given by B(X,Y) = Tr (ad X)(ad Y), then is a real inner product on . Under the adjoint representation, K is the subgroup of G that preserves this inner product. If H is another compact subgroup of G, then averaging the inner product over H with respect to the Haar measure gives an inner product invariant under H. The operators Ad p with p in P are positive symmetric operators. This new inner produst can be written as where S is a positive symmetric operator on such that Ad(h)tS Ad h = S for h in H (with the transposes computed with respect to the inner product). Moreover, for x in G, So for h in H, For X in define If ei is an orthonormal basis of eigenvectors for S