Is every connected semisimple linear Lie group the identity connected component of (the real points of) an algebraic group?

I was told some fact along this line is true but could not find any reference after searching for a while.

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Is every connected semisimple linear Lie group the identity connected component of (the real points of) an algebraic group?

I was told some fact along this line is true but could not find any reference after searching for a while.

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(Comments converted into an answer:)

Mostow (to whom I think this is often attributed?) gives a detailed proof in (1949, Lemmas 2.2, 2.3).

Borel (2001, pp. 152, 114) notes that algebraicity of perfect (e.g. semisimple) linear Lie algebras was claimed by Cartan in (1897, p. 547), and spells out what “it may not too far fetched to believe” would have been his simple proof.

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