Geodesic
Quotient spaces modulo tori and transitive actions of Lie groups
Sbornik. Mathematics, Tome 186 (1995) no. 9, pp. 1375-1388

Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that a transitive locally effective action of a compact connected simply-connected Lie group on a manifold of the form $K/A$, where $K$ is a non-simple compact connected simply-connected Lie group all of whose simple factors have rank greater than 5 and $A$ is a torus in $K$ in general position of corank 1, is equivalent to the original action of the group $K$. 
@article{SM_1995_186_9_a9,
     author = {A. N. Shchetinin},
     title = {Quotient spaces modulo tori and transitive actions of {Lie} groups},
     journal = {Sbornik. Mathematics},
     pages = {1375--1388},
     publisher = {mathdoc},
     volume = {186},
     number = {9},
     year = {1995},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1995_186_9_a9/}
}
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A. N. Shchetinin. Quotient spaces modulo tori and transitive actions of Lie groups. Sbornik. Mathematics, Tome 186 (1995) no. 9, pp. 1375-1388. http://geodesic.mathdoc.fr/item/SM_1995_186_9_a9/