The paper contains well-known classifications of 3- and 4-dimensional simply connected Riemannian manifolds with transitive isometry groups. The classification is known since the fifties [see e.g. S. Ishihara, J. Math. Soc. Japan 7, 345-370 (1955; Zbl 0067.39602)].
However, the author obtains this classification by means of a new technique of Cartan triples. An \(n\)-dimensional Cartan triple is a triple \(({\mathfrak g},\Gamma,\overline{\Omega})\), where \(\mathfrak g\) is a Lie subalgebra of \({\mathfrak O}(n)\), \(\Gamma:\mathbb{R}^n\to{\mathfrak g}^\perp\) is a linear map and \(\overline{\Omega}:\mathbb{R}^n\times \mathbb{R}^n\to{\mathfrak g}\) is a bilinear skew map, which are both \(\mathfrak g\)-invariant and satisfy some additional identities involving their associated A-S torsion and curvature \(T\) and \(\widetilde{\Omega}\). This torsion and curvature are defined as follows: \[ T(X,Y)=\Gamma(Y)X-\Gamma(X)Y,\quad \widetilde{\Omega}(X,Y)=\overline{\Omega}(X,Y)-[\Gamma(X),\;\Gamma(Y)]_{\mathfrak g}. \] The classification is obtained by establishing a correspondence between Cartan triples and Riemannian homogeneous spaces. This correspondence is explained in the article, but for the proof the reader is referred to [V. Patrangenaru, Geom. Dedicata 50, No. 2, 143-164 (1994)].