Let $k$ be a field of characteristic zero and let $G$ be an adjoint semi-simple algebraic group over $k$.


On p34 of the paper "Sansuc - Groupe de Brauer et arithmétique des groupes
algébriques lineaires sur um corps de nombres", it is claimed that there exists a collection of finite field extensions $k \subset k\_i$ such that
$$G \cong \prod\_i \mathrm{R}\_{k\_i/k}(G\_i) \quad (\*)$$
where the $G\_i$ are absolutely simple adjoint groups over $k\_i$ and $\mathrm{R}\_{k\_i/k}$ denotes the Weil restriction.


I was quite surprised when I saw this, as it certainly seems to be something special about adjoint groups. However, Sansuc unfortunately gives no explanation nor reference why this holds.



> 
> Why does the stated isomorphism (\*) exist?
> 
> 
> 


I would be happy with either a proof or a reference.



Adjoint semi-simple algebraic groups over non-algebraically closed fields
See Proposition 6.4.4 and Remark 6.4.5 in Brian Conrad's article "Reductive groups schemes" in "Autour des schemas en groupes, Vol. I" (alternatively, <http://math.stanford.edu/~conrad/papers/luminysga3smf.pdf>) for a proof (in a more general setting). 


As you mention, this is special for adjoint groups (but would also work for simply connected groups).


