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 $G_0\subset G$ 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