Let $G$ be a connected reductive algebraic group over a local field $F$ with fixed maximal torus $T$, and denote by $R = R(G,T)$ the set of roots of $T$ in $G$, namely, the set of all nontrivial characters $\alpha$ of $T$ such that the space $$ \mathfrak{g}_\alpha := \{ Y \in \mathfrak{g} : \operatorname{Ad}(t)Y = \alpha(t)Y \text{ for all } t \in T \} $$ is nonzero, where $\mathfrak{g}$ is the Lie algebra of $G$. It is a basic result that $R$ is a root system in the abstract sense, which is reduced if $T$ is split.
I want to know if there is a correspondence between the set of subsystems of $R(G,T)$ that are themselves root systems, and the set of Levi subgroups of $G$ that contain $T$? The inclusion one way is obvious, but it is not clear to me if it is possible to construct a Levi subgroup to correspond to a given subsystem of $R$.
EDIT: I should add a clarifying point. Put $S \subset T$ for the maximal $F$-split subtorus of $T$, and denote by $X$ (respectively, ${_FX}$) the lattice of characters of $T$ (respectively, $S$).
It is known that parabolic subgroups containing a fixed minimal parabolic subgroup are in one-to-one correspondence with the nonzero elements in the image of the map $$ \pi : \mathbf{R} \otimes_{\mathbf{Z}} X \to \mathbf{R} \otimes_{\mathbf{Z}} ({_FX}) $$ induced by restriction of characters from $T$ to $S$, which in turn is a base of the root system ${_FR}$ of $S$ in $G$.
As such, I really should expect the correspondence I want to come from subsystems of ${_FR}$, not $R$ itself (although of course the systems are related).
Of course, this difference only arises if $F$ is not algebraically closed and $T$ is not $F$-split.