Let $ G(\mathbb{R}) $ be a noncompact Lie group which is the real points of an algebraic group. Let $ \Gamma $ be a closed subgroup of $ G $.
$ \Gamma $ is called Lie primitive if it is not contained in any proper closed positive dimensional subgroup of $ G $.
Is it true that a subgroup $ \Gamma $ is Lie primitive if and only if it is a discrete Zariski dense subgroup of $ G(\mathbb{R}) $?
I'm asking this in response the comment here
about thin groups.