Let $ G $ be a simple Lie group and let $ \Gamma $ be a lattice in $ G $. If $ \Gamma $ is not contained in any connected subgroup of $ G $ does that imply that $ \Gamma $ is not contained in any positive dimensional subgroup of $ G $?
As pointed out in the comments, an equivalent way of saying this is: If $ \Gamma $ is not contained in any connected subgroup of $ G $ does that imply that the only Lie subalgebras $ \mathfrak{h} \subset \mathfrak{g} $ normalized by $ \Gamma $ are $ \mathfrak{h}=0 $ and $ \mathfrak{h}=\mathfrak{g} $?
Context: I was thinking about the lattice $ SL(2,\mathbb{Z}) $ in $ SL(2,\mathbb{C}) $. $ SL(2,\mathbb{Z}) $ is not contained in any proper connected subgroup of $ SL(2,\mathbb{C}) $. And it is also true that $ SL(2,\mathbb{Z}) $ is not contained in any positive dimensional proper subgroup.
Cross-posted on Math.se: it has been up on MSE for over a week with 1 upvote and no comments or answers so I thought MO might be better.
