How Can I Tell when A Subgroup of a Lie Group is Generated by Unipotents? I'm trying to understand the proof of the Oppenheim conjecture using Ratner's theorem, and I don't immediately see why $SO(2,1)$ is generated by unipotents.  Why is $SO(2,1)$ generated by unipotents?  More generally, are there equivalent conditions to being generated by unipotents that are easier to check?
 A: Groups like SO$(2,1)$ have been studied in several frameworks: geometry and generation of classical groups over various ground fields, where unipotent elements tend to appear as transvections; real Lie groups, where the structure theory of groups like this is well developed (as in Helgason's old book, for example, republished by AMS); real points of (almost) simple algebraic groups, where papers by Borel and Tits have worked out the abstract group structure and generation in considerable generality.    From the last point of view, I guess the crucial point about your question is that the group is isotropic over $\mathbb{R}$ and thus noncompact as a Lie group unlike SO(3).   In particular, it has nontrivial unipotent elements (here of infinite order as group elements); the subgroup they generate in the algebraic group setting is closed and normal as
well as defined over $\mathbb{R}$.   
Your example is old and references can be quoted, but you need to keep in mind the different approaches possible.   What is simplest here depends a lot on your viewpoint.
P.S. The blog entry by Terry Tao suggested by Jack Schmidt gives a nice way to visualize the 3-dimensional situation, but the general perspectives I've sketched are useful in higher dimensions.
A: For $SO(2,1)$ the problem is easy, since the group is isomorphic to $SL(2,\RR)$. To see this consider the action by conjugation of $SL(2,R)$ on the vector space of $2 \times 2$ matrices with real coefficients and trace zero. This action preserves determinant, which is a quadratic form of signature $(2,1)$. This gives you an isomorphism. Then all you need to do is to track the obvious unipotents in $SL(2)$ and see their image in $SO(2,1)$.
In the context of Oppenheim conjecture, you can also refer to any one of the early papers of
Margulis. He explicitly described the required unipotents.
