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I suspect "compact Lie group" should read "simple compact Lie group" (or "semisimple" or "reductive", as these are trivial consequences) as all cases listed are simple Lie groups, but one needs to double-check the reference: maybe the theorem proves that non-simple Lie groups cannot have such metrics. — Preceding unsigned comment added by 140.105.42.135 (talk) 23:03, 11 August 2016 (UTC)