Question: What are the connected subgroups of $SL(2,\mathbb{R})$?
Connected Lie subgroups of $SL(2,\mathbb{R})$ are in 1:1-correspondence with Lie subalgebras of $\mathfrak{sl}(2,\mathbb{R})$. If we denote the usual root decomposition of $\mathfrak{sl}(2,\mathbb{R})$ by
$$ \mathfrak{sl}(2,\mathbb{R}) = \mathfrak{g}_0 \oplus \mathfrak{g}_\alpha \oplus \mathfrak{g}_{-\alpha}, $$
then we clearly have the following Lie subalgebras: $\{0\}, \mathfrak{g}_0, \mathfrak{g}_\alpha, \mathfrak{g}_{-\alpha}, \mathfrak{so}(2), \mathfrak{g}_0 \oplus \mathfrak{g}_\alpha, \mathfrak{sl}(2,\mathbb{R})$ and all conjugates of those, e.g. $Ad(g)(\mathfrak{g}_\alpha)$ for $g \in SL(2, \mathbb{R})$.
But I have a hard time imagining what other groups are there.
Background: I want to know all subgroups of the isometry group of $Isom(\mathbb{RH}^2 \times \mathbb{RH}^2)=SL(2,\mathbb{R}) \times SL(2,\mathbb{R})$. That classification will follow with Goursat's lemma from the answer to my question.