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...



If split algebraic groups are potentially isomorphic, are they isomorphic? 
The answer is yes, for arbitrary split connected reductive groups over any field. The main point is that the Existence, Isomorphism, and Isogeny Theorems (relating split connected reductive groups and root data) are valid over any field. One reference is SGA3 near the end (which works over any base scheme), but in Appendix A.4 of the book "Pseudo-reductive groups" there is given a direct proof over fields via faithfully flat descent, taking as input the results over algebraically closed fields (since for some reason the non-SGA3 references always seem to make this restriction). 


[Caveat: that A.4 gives a complete treatment for the Isomorphism and Isogeny Theorems over general ground fields, and that is what the question is really about anyway; for the Existence Theorem in the case of exceptional types I don't know a way to "pull it down" from an algebraic closure, instead of having to revisit the constructions to make them work over prime fields or $\mathbf{Z}$.]


