The Structure of Cartan Subgroups in Lie Groups

We study properties and the structure of Cartan subgroups in a connected Lie group. We obtain a characterisation of Cartan subgroups which generalises W\"ustner's structure theorem for the same. We show that Cartan subgroups are same as those of the centralizers of maximal compact subgroups of the radical. Moreover, we describe a recipe for constructing Cartan subgroups containing certain nilpotent subgroups in a connected solvable Lie group. We characterise the Cartan subgroups in the quotient group modulo a closed normal subgroup as the images of the Cartan subgroups in the ambient group. We also study the density of the images of power maps on a connected Lie group and show that the image of any $k$-th power map has dense image if its restriction to a closed normal subgroup and the corresponding map on the quotient group have dense images.