Let $G$ be a compact connected semisimple Lie group and let $\mathfrak g$ denote its Lie algebra. Is the following result true? Does it follows directly from Dynkin's classification of maximal Lie subalgebras?:

There are $\mathfrak h_1,\dots,\mathfrak h_r$ proper Lie subalgebras of $\mathfrak g$ such that: (1) for any Lie subalgebra $\mathfrak a$ of $\mathfrak g$ there is $a\in G$ such that $\textrm{Ad}_a(\mathfrak a)\subset \mathfrak h_i$ for some $i$; (2) if a Lie subalgebra $\mathfrak a$ of $\mathfrak g$ contains properly $\mathfrak h_i$ for some $i$, 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 in $G$.

I am confident that the answer is affirmative, except in the finiteness of the number of Lie subalgebras $\mathfrak h_i$. I hope there is a precise (modern) reference for it.

Some related questions:

Modern reference for maximal connected subgroups of compact Lie groups