Reductive subgroup corresponding to a subdiagram of the Dynkin diagram of a simple group

I am looking for a reference for the following well-known fact: For any subdiagram $Delta_0$ of of the Dynkin diagram $Delta=D(G)$
of an adjoint simple group $G$ over an algebraically closed field $k$, there exists a reductive subgroup of maximal rank $G_0subset G$
with Dynkin diagram $Delta_0$.

To be more precise, I am looking for a reference for a proof of the following well-known lemma:

Lemma 1. Let $G$ be an adjoint, connected, simple algebraic group with Dynkin diagram $Delta=D(G)$
over an algebraically closed field $k$ of any characteristic.
Let $Delta_0$ be a subdiagram of $Delta$
(that is, a subset $Pi_0$ of the set $Pi$ of vertices of $Delta$,
together with all the edges of $Delta$ connecting pairs of vertices of $Pi_0$).
Then there exists a connected reductive $k$-subgroup of maximal rank $G_0$ of $G$
such that the corresponding adjoint semisimple group $G_0^{ad}$
has Dynkin diagram $Delta_0$.

I know a simple proof of Lemma 1, but I would prefer to give a reference rather than a proof.

The proof goes as follows. Let $T$ be a maximal torus of $G$, and let $R=R(G,T)$ be the root system,
then our $Pi$ is a basis of $R$. Let $S$ be the subgroup of $T$ orthogonal to $Pi_0$,
then it is a subtorus of $T$ (because $G$ is adjoint).
Set $G_0=C_G(S)$, the centralizer of $S$ in $G$.
Then $G_0$ is a connected reductive subgroup of $G$.
It is easy to see that (the adjoint group of) $G_0$ has Dynkin diagram $Delta_0$.

Note that Lemma 1 is a special case of the following Lemma 2,
for which I would also be happy to have a reference.

Lemma 2. Let $G$ be an adjoint, connected, simple algebraic group
over an algebraically closed field $k$ of any characteristic.
Let $T$ be a maximal torus of $G$, and let $R=R(G,T)$ be the root system.
Let $R_0$ be a closed symmetric subset of $R$.
Then there exists a connected reductive $k$-subgroup of maximal rank $G_0$ of $G$
with root system $R_0$.

I will be grateful to any references, comments, etc. (also to a proof of Lemma 2).

Mikhail Borovoi