Geodesic
The existence of maximal invariant subalgebras in simple Cartan-type Lie algebras
Matematičeskie zametki, Tome 9 (1971) no. 2, pp. 211-222.

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It is proved that a Cartan-type Lie algebra $\mathfrak{L}$ contains a maximal subalgebra, invariant with respect to the group of automorphisms of $\mathfrak{L}$. 
@article{MZM_1971_9_2_a11,
     author = {V. A. Kreknin},
     title = {The existence of maximal invariant subalgebras in simple {Cartan-type} {Lie} algebras},
     journal = {Matemati\v{c}eskie zametki},
     pages = {211--222},
     publisher = {mathdoc},
     volume = {9},
     number = {2},
     year = {1971},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1971_9_2_a11/}
}
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V. A. Kreknin. The existence of maximal invariant subalgebras in simple Cartan-type Lie algebras. Matematičeskie zametki, Tome 9 (1971) no. 2, pp. 211-222. http://geodesic.mathdoc.fr/item/MZM_1971_9_2_a11/