It is expected that a `small? odd type infinite simple group of finite Morley rank is isomorphic to the Chevalley group $PGL_2(K)$ for $K$ algebraically closed. We discuss different set-ups with the common feature that a group of finite Morley rank $G$ is a definable closure of a group $PGL_2(F)$ (or $PSL_2(F)$) for some infinite field $F$. I will explain what kind of assumptions are needed for the identification of $G$ with $PGL_2(K)$ in such set-ups. I will also present a tool, particularly suitable for our set-up, for recognising a connected odd type group of finite Morley rank with $PGL_2(K)$. Some parts of the talk are joint work with P. Ugurlu and some work in progress with A. Deloro.