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Jul 15, 2017 at 22:35 comment added LSpice I guess you want to specify in (1) that $K$ has odd characteristic (which is easily arranged).
Dec 21, 2012 at 18:31 comment added David Corwin I actually really like the first proof and have thought about Galois-theoretic ways to prove (or at least think about how to prove) group-theoretic results.
Oct 13, 2011 at 7:04 comment added Mariano Suárez-Álvarez The second one is actually a great, non-awfully sophisticated argument :) It is essentially the same as the purely algebraic proof of complete reducibility of finite dimensional modules over a fin. dim. semisimple Lie algebra---in that case, one replaces the coprimality by the action of the Casimir element, a completely parallel argument.
Oct 13, 2011 at 6:42 history answered Matthias Künzer CC BY-SA 3.0