Let $G$ be an algebraic group over a field $k$ of characteristic zero. My understanding is that solvability is not a geometric property (is this correct though?). This motivates my questions:
Let $K/k$ be a finite extension. Suppose $G$ is solvable. Is $G_K$ still solvable? What about when $K = \bar{k}$?
Conversely,
If $G_{\bar{k}}$ is solvable, does this imply that $G$ is solvable?