  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.
