Reference request, relation algebraic groups, real groups and their Lie algebras

Question

Let's work in characteristic $0$. Let $G$ be a semi-simple algebraic group defined over a subfield of $\mathbb{R}$ (invariant under transposition). Let $\mathfrak{g}$ be its Lie algebra (Zariski-tangent space).

The following are well known:

1. The $\mathbb{R}$-points $G_{\mathbb{R}}$ of $G$ is a real Lie group.
2. The $\mathbb{R}$-points $\mathfrak{g}_{\mathbb{R}}$ of $\mathfrak{g}$ is a real Lie algebra.
3. $\mathfrak{g}_{\mathbb{R}}$ the Lie algebra of the Lie group $G_{\mathbb{R}}$.

Is there a reference for this fact? I am also interested in other correspondences, such as the following: if $S\subseteq G$ is a maximal ($\mathbb{K}$-split) torus, where can I find that the relative root system of the algebraic group corresponds to the root system of the real Lie group with respect to the abelian subgroup $S_{\mathbb{R}}$?