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> Is every connected semisimple linear Lie group the identity connected component of (the real points of) an algebraic group?
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I was told some fact along this line is true but could not find any reference after searching for a while.



Is every connected semisimple linear Lie group the connected component of (the real points of) an algebraic group?
(Comments converted into an answer:)


Mostow (to whom I think this is often attributed?) gives a detailed proof in ([1949](//ams.org/mathscinet-getitem?mr=32656), Lemmas 2.2, 2.3).


Borel ([2001](//ams.org/mathscinet-getitem?mr=2002g:01010), pp. 152, 114) notes that algebraicity of perfect (e.g. semisimple) linear Lie algebras was claimed by Cartan in ([1897](//zbmath.org/?q=an:26.0174.02), p. 547), and spells out what “it may not too far fetched to believe” would have been his simple proof.


