Suppose $K$ is a (not necessarily algebraically closed) field, and $G_1$ and $G_2$ are *split* semisimple algebraic groups over $K$ which become isomorphic over $\bar{K}$, the algebraic closure of $K$. Are $G_1$ and $G_2$ isomorphic over $K$? What about if the $G$s are reductive?

It seems like this should follow (at least in the semisimple case) from Tits' general structure theorem for semisimple groups over a not-necessarily algebraically closed field; as explained in section 35.5 of Humphreys' *Linear Algebraic Groups*, a semisimple algebraic group is determined by its $\bar{K}$ isomorphism class, its anisotropic kernel (which looks like it is trivial for a split group) and its `index' (for which there again only seems to be one choice for a split group). But I am not expert enough to completely trust this argument...