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In general, a semisimple algebraic group is an almost-direct product of almost-simple groups. However for simply connected semisimple algebraic groups, it holds that they are a direct product of simply connected simple algebraic groups.

I tried to find a proof for this, but unfortunately on the internet everybody just states this fact without proof! Does someone maybe have a reference for this?

Kind regards

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    $\begingroup$ For a proof, see Proposition $1.4.10$ here. $\endgroup$ – Dietrich Burde Sep 2 '16 at 19:53
  • $\begingroup$ the tag abstract-algebra should be replaced with algebraic-groups $\endgroup$ – YCor Sep 4 '16 at 8:14

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