How can one describe the orbits of the Lie group $G=\mathrm{SU}(2,1)$ in its Lie algebra $\mathfrak{g}=\mathfrak{su}(2,1)$ with respect to the adjoint representation? 



Orbits in the adjoint representation of $SU(2,1)$
A systematic method to do this for any real reductive group is given in Propositions 2.10 through 2.13 of Vogan's course [*The method of coadjoint orbits for real reductive groups*](http://www.ams.org/mathscinet-getitem?mr=1737729) (available from [his web page](http://www-math.mit.edu/~dav/paper.html)). As he says, you'll also need [Collingwood-McGovern](http://www.ams.org/mathscinet-getitem?mr=1251060) for the last step things eventually boil down to: classifying nilpotent orbits of (smaller) reductive groups.


Beyond classification (i.e. describing $\mathfrak g/G$) Vogan also *describes* to some extent the structure of each orbit, if that's what you meant.


(An earlier paper of [Burgoyne-Cushman](http://www.ams.org/mathscinet-getitem?mr=432778) gives another, to me less transparent method.)


