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accepted Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer?
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revised Properties of the longest element in a Weyl group
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revised Properties of the longest element in a Weyl group
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answered Properties of the longest element in a Weyl group
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comment Direct proof to show that a set is closed
Your proof is a beautiful one-liner. Why would you want something else? A "more direct" proof will likely just explicitly reprove the facts you've used.
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comment A question on sequence of functions that diverges everywhere
note: cross-posted at mathoverflow.net/questions/67668
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comment Does $G\oplus G \cong H\oplus H$ imply $G\cong H$ in general?
Thanks. Googling around, it looks like the OP's question is one of "Kaplansky's test problems" which people like to ask in various categories. Another one is, "If $G$ and $H$ are direct summands of each other, are they isomorphic?" The $G$ produced in Corner's paper obviously answers this problem in the negative for abelian groups.