Suppose H and K are algebraic subgroups of a linear algebraic group G. Suppose that G acts on a smooth algebraic variety X.
Given $x\in X$, we can consider the map \begin{equation} H\times K \rightarrow X, \quad (h,k)\mapsto hkx. \end{equation} If HK happens to be a group, then the image is locally closed (it's an orbit). But what if HK is not a group? Is there an example of an H, K, X where the image is only constructible?