We describe a straightforward strategy that can be used to construct dense embeddings of certain amalgamated products into simple linear Lie groups. The geometric nature of the strategy allows one to dictate properties of the output dense embedding; for instance, it is sometimes possible to ensure a lack (or presence) of nontrivial unipotents in the image, or arrange for each matrix in the image to have algebraic entries. This is joint work in progress with Konstantinos Tsouvalas.
