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If a Lie algebra is semisimple or reductive, its Cartan subalgebras are Abelian, and their elements semisimple.

Are there non-reductive algebras with Abelian Cartan subalgebras all of whose elements are semisimple?

N.B.: I asked this also on MO.

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Does the same not hold for any reductive Lie algebra, as we can then get a Cartan as the direct sum of a cartan of the derived algebra and the center? – Tobias Kildetoft Feb 17 '12 at 10:07
@Tobias, ah indeed! I actually meant reductive. Fixed! – Mariano Suárez-Alvarez Feb 17 '12 at 18:56

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