Subgroups of compact Lie groups generated by a subset of nodes of the Dynkin diagram

Question

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.

Answer 1

A root defines in your Lie algebra a non zero element in you algebra up to multiplication by a scalar. A collections of roots defines a subvector space in you Lie algebra.

A set of simple roots defines a set of root obtained by linear combinaisons of them with coefficients in $\mathbb Z$. You then obtain a linear subspace of your Lie-algebra. It is not a sub Lie algebra as its intersection with the Cartan algebra $\mathfrak t$ is zero.

You have a minimal and a maximal choice here. You can add the whole $\mathfrak t$ to obtain $\mathfrak g_1$ or just the linear span of the coroots associated to your chosen roots. The second option amount to take the Lie closure $\mathfrak g_2$ of your subspace. The subalgebra $\mathfrak g_1$ is the direct sum of $\mathfrak g_2$ and some abelian algebra $\mathfrak h_0$ of dimension $r - |I|$. The orthogonal $\mathfrak h_1$ of $\mathfrak h_0$ in $\mathfrak h$ is a Cartan subalgebra of $\mathfrak g_1$ and your choosen subset of roots is the set of roots for $(\mathfrak g_1, \mathfrak h_1)$.

Exponentiate the real part of this algebra in your group $G$ to obtain a group with Lie algebra $\mathfrak g_1$ or $\mathfrak g_2$.