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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Mariano Suárez-Álvarez
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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
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@Tobias, ah indeed! I actually meant reductive. Fixed! – Mariano Suárez-Álvarez Feb 17 '12 at 18:56