Where can I find a reference for the following fact, or as close as possible to it?

Let $G$ be a semisimple compact real Lie group with rank $r$, let $T$ be a maximal torus in $G$, let $\mathfrak{g}$ and $\mathfrak{t}$ be their Lie algebras, let $\Phi$ be the root system of $\mathfrak{g}$ relative to $\mathfrak{t}$, and let $\alpha_1,\ldots,\alpha_r \in \Phi$ be a basis of simple roots. Then for any subset $I \subseteq \{1,\ldots,r\}$, the subgroup $G_I$ of $G$ associated to the Lie subalgebra of $\mathfrak{g}$ whose complexification is generated by $\mathfrak{t}$ and by the coroots $\alpha_i^\vee$ for $i\in I$, is (isogenous to?) the product of a torus of rank $r - \#I$ and a semisimple group (of rank $\#I$) described by the Dynkin diagram consisting of those nodes of the Dynkin diagram of $\Phi$ labeled by an element of $I$.

Also, how can I state this, or describe the subgroup $G_I$, in a less clumsy fashion? (Hopefully avoiding such phrases as “the maximal compact subgroup of the Levi factor of the parabolic subgroup defined by $I$ in the complexification”.)

I am looking for a description and discussion of the situation that stays as much as possible within the context of *compact* real Lie groups. (Essentially, how do I convince someone with little background in Lie or algebraic groups, that a subset $I$ of nodes of the Dynkin diagram of $G$ defines a subgroup of $G$ “as expected”?)

**Edit:** Let me emphasize that I'm looking for a reference, not a proof. The closest I found so far is §7–8 in chapter V of Knapp's *Lie Groups Beyond an Introduction* or proposition 12.6 in Malle & Testerman's *Linear Algebraic Groups and Finite Groups of Lie Type*, but they remain annoyingly far from the above statement.

the intersection with $G$ of the parabolic(...)? $\endgroup$1more comment