Assume you have an almost connected simple Lie group G with trivial center. (In particular excluding non-algebraic examples such as the universal cover of SL\_2(R).)


Such a group should automatically be an algebraic group over the reals resp. the complex numbers. 


Is this true and why? 


Can we in addition conclude (EDIT: under a good choice of the field and possibly additional assumptions?) that G is absolutely almost simple as an algebraic group? 


EDIT: Asking this I do not want to regard a complex Lie group as a real algebraic group. 



Are certain simple Lie groups linear algebraic groups?
The answer is yes for complex Lie groups, and follows from the classification. (Root data are in fact defined over $\mathbb{Z}$: a complex semisimple group has not only an underlying algebraic, but even an arithmetic structure). For a more direct explanation, see Theorem 6.3 in the book "Lie Groups and Lie Algebras III" by Onischik-Vinberg: any connected complex Lie group satisfying $G = [G,G]$ and admitting a faithful linear representation (which for semisimple groups is automatic), has a unique underlying complex algebraic structure.


For real Lie groups this is not *quite* true, as noted above.


