Let $ G $ to be a semi-simple Lie group connected compact and let $ mathfrak g $ denote his Lie algebra.

Does Dynkin's classification of maximal Lie subalgebras imply the following result?

There is $ mathfrak h_1, dots, mathfrak h_r $ Lie subalgebras of $ mathfrak g $ such as: (1) for any Lie subalgebra $ mathfrak a $ of $ mathfrak g $ there is $ in G $ such as $ textrm {Ad} _a ( mathfrak a) subset mathfrak h_i $ for some people $ i $;

(2) if a Lie subalgebra $ mathfrak a $ of $ mathfrak g $ contains correctly $ mathfrak h_i $ for some people $ i $then $ mathfrak a = mathfrak g $;

(3) for each $ i $, the only connected Lie subgroup $ H_i $ of $ G $ with Lie algebra $ mathfrak h_i $ is closed $ G $.

I am satisfied that the answer is affirmative except for the number of Lie subalgebras. $ mathfrak h_i $.

I hope that there is a precise (modern) reference for this.

Some related questions:

Modern Reference for Maximum Connected Subgroups of Compact Lie Groups

Maximum subgroups of semi-simple Lie groups