In a joint paper with Yifan Yang we constructed an "exotic" embedding of $SL_2(\mathbb R)$ in $Sp_4(\mathbb R)$ (in fact, of $PSL_2(\mathbb R)$ in $PSp_4(\mathbb R)$), namely, $$ \iota\colon\begin{pmatrix} a & b \cr c & d \end{pmatrix} \mapsto\begin{pmatrix} a^2d+2abc & -3a^2c & abd+\frac12b^2c & \frac12b^2d \cr -a^2b & a^3 & -\frac12ab^2 & -\frac16b^3 \cr 4acd+2bc^2 & -6ac^2 & ad^2+2bcd & bd^2 \cr 6c^2d & -6c^3 & 3cd^2 & d^3 \end{pmatrix}. $$ An equivalent form of the embedding was independently discovered by Don Zagier, and we could not find it in the literature.

Although the properties of the embedding (discussed in the preprint above) are nice by themselves, I am interested in an exhaustive list of possibilities to embed other matrix groups and their direct products in $Sp_4(\mathbb R)$ (or $PSp_4(\mathbb R)$). For example, can the direct product of two copies of $SL_2(\mathbb R)$ be embedded?

As I am not a specialist in Lie groups, I would appreciate plainer sources. Thank you for any help in advance!

thinkit makes the standard parabolic subgroups a bit nicer-looking? Of course you can easily move from one form to the other via some conjugation. I don't know if Tilouine would know too much about these exotic SL_2's though. – Kevin Buzzard Jul 5 '10 at 12:08