Coadjoint Orbits of Lie Algebras and Cartan Class

We study the coadjoint orbits of a Lie algebra in terms of Cartan class. In fact, the tangent space to a coadjoint orbit $\mathcal{O}(\alpha)$ at the point $\alpha$ corresponds to the characteristic space associated to the left invariant form $\alpha$ and its dimension is the even part of the Cartan class of $\alpha$. We apply this remark to determine Lie algebras such that all the nontrivial orbits (nonreduced to a point) have the same dimension, in particular when this dimension is $2$ or $4$. We determine also the Lie algebras of dimension $2n$ or $2n+1$ having an orbit of dimension $2n$.