TITLE: Which maximal closed subgroups of Lie groups are maximal subgroups?
QUESTION [6 upvotes]: Which maximal closed subgroups of Lie groups are maximal subgroups?

REPLY [2 votes]: There is a paper by M. Golubitsky, "Primitive actions and maximal subgroups of Lie groups", J. Differ. Geom. 7 (1972), 175-191: from the Introduction: "...there exist maximal Lie subgroups whose Lie algebras are not maximal subalgebras".