Standard Levi subgroups of a split reductive group

$L_I=C_G(T_I)$ is smooth connected reductive by [F1]; its Lie algebra is $\mathfrak{t}\oplus {⨁}_{\alpha \in {\mathrm{\Phi }}_I}{\mathfrak{g}}_{\alpha }$ because the characters vanishing on $T_I$ lie in the rational span of $I$. Each root has integral coefficients in the simple basis, so a root in this rational span has zero coefficients outside $I$ and lies in $\mathbb{Z}I\cap \mathrm{\Phi }$. Hence the roots of $(L_I,T)$ are precisely the elements of ${\mathrm{\Phi }}_I$, and the corresponding root groups are $U_{\alpha }\subseteq L_I$; the coroot assignment restricts, and the Weyl group of $(L_I,T)$ is the subgroup $W_I$ generated by the $s_{\alpha }$ with $\alpha \in I$ (these are the reflections in the simple roots of the subsystem). This proves (a).

$T_I$ is the largest subtorus of $T$ killed by all $\alpha \in I$ because any such subtorus lies in ${\bigcap }_{\alpha \in I}\ker \alpha$ and, being a torus, in its maximal reduced subtorus; the character-module quotient by its torsion defines exactly that subtorus. By (a) and [F2] applied to $L_I$, the group $L_I$ is generated by $T$ and the root groups $U_{\alpha }$ with $\alpha \in {\mathrm{\Phi }}_I$; this proves (b). For (c): $B\cap L_I$ is a Borel subgroup of $L_I$ by the centralizer-intersection input [F4]; its Lie algebra is $\mathfrak{t}\oplus {⨁}_{\alpha \in {\mathrm{\Phi }}_I\cap {\mathrm{\Phi }}^{+}(B)}{\mathfrak{g}}_{\alpha }$, so its positive system is ${\mathrm{\Phi }}_I\cap {\mathrm{\Phi }}^{+}(B)$, and the simple roots of the subsystem relative to this positive system are exactly the elements of $I$ (a simple root of $\mathrm{\Phi }$ lying in ${\mathrm{\Phi }}_I$ is indecomposable in ${\mathrm{\Phi }}_I$, and every root of ${\mathrm{\Phi }}_I$ is a nonnegative combination of $I$).

Fix $\alpha \in \mathrm{\Delta }$. Finite linear algebra supplies ${\eta }_{\alpha }$ with the given simple-root pairings; clearing denominators gives integral ${\lambda }_{\alpha }=m_{\alpha }{\eta }_{\alpha }$, $m_{\alpha }>0$. For $\beta \mathrm{\ne }\alpha$ one has $⟨\beta ,{\lambda }_{\alpha }⟩=0$, so both $U_{\beta }$ and $U_{-\beta }$ and the torus $T$ centralize this cocharacter. They generate $G_{\beta }$ by [F2], giving $G_{\beta }\subseteq Z_G({\lambda }_{\alpha })\subseteq P_G({\lambda }_{\alpha })$. The dual reflection formula gives $s_{\beta }({\lambda }_{\alpha })={\lambda }_{\alpha }$; hence every representative of $s_{\beta }$ fixes ${\lambda }_{\alpha }$ by conjugation and lies in its centralizer, proving(d). The simple root $\alpha$ itself pairs positively, and its negative root group is excluded; no representative of $s_{\alpha }$ is claimed to lie in this parabolic. Central changes of a rational dual vector pair trivially with every root and do not affect these limit subgroups.