Which maximal closed subgroups of Lie groups are maximal subgroups?

3$\begingroup$ What's an example of a maximal closed subgroup of a nondiscrete Lie group? $\endgroup$ – Pete L. Clark May 18 '11 at 19:56

$\begingroup$ Pete: See arxiv.org/PS_cache/math/pdf/0605/0605784v3.pdf . $\endgroup$ – David Feldman May 18 '11 at 21:50

$\begingroup$ For noncompact Lie groups it's probably essential in this question to separate the semisimple (or reductive) ones from the others, since the compact group preprint mentioned here and its many references indicate already the richness of the problem in the compact case where the structure theory of semisimple groups predominates. In general, there is also a need to compare real and complex Lie groups. But at least in the semisimple (or reductive) case, parallel work on algebraic groups should be a helpful guide. Is there any relevant literature on solvable Lie groups? $\endgroup$ – Jim Humphreys May 18 '11 at 22:14

$\begingroup$ @David: thanks, that was helpful. I worked through the commutative case in my head and saw that that was bad for maximal subgroups. I should have thought about it more: I do know that every compact subgroup of $\operatorname{GL}_n(\mathbb{R})$ is contained in an orthogonal group... $\endgroup$ – Pete L. Clark May 19 '11 at 7:50

$\begingroup$ A related question is mathoverflow.net/questions/60315/… $\endgroup$ – Alain Valette Aug 8 '11 at 8:17
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".