Which maximal closed subgroups of Lie groups are maximal subgroups?
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There is a paper by M. Golubitsky, "Primitive actions and maximal subgroups of Lie groups", J. Differ. Geom. 7 (1972), 175191: from the Introduction: "...there exist maximal Lie subgroups whose Lie algebras are not maximal subalgebras". 

