It seems to me every book has a different definition for reductive Lie groups. I've never seen anyone use the definition which seems most natural: a Lie group whose tangent algebra is reductive (i.e. a direct sum of an abelian and semisimple Lie algebras). Doesn't anyone discuss such groups?
More importantly and more concretely, is there a connected Lie group G whose tangent algebra is reductive but whose commutator is not closed?