Roots and root groups of a split reductive group

Let $(G,T)$ be a split reductive group over $k$ (Split reductive groups). The roots $\mathrm{\Phi }(G,T)$ are the nontrivial characters $\alpha \in X(T)$ for which the adjoint weight space ${\mathfrak{g}}_{\alpha }$ in $\mathfrak{g}=\mathrm{Lie}G$ is nonzero. The adjoint action is rational, and the weight decomposition $\mathfrak{g}={\mathfrak{g}}_0\oplus {⨁}_{\alpha \in \mathrm{\Phi }(G,T)}{\mathfrak{g}}_{\alpha }$ is the choice-free decomposition into character eigenspaces (Representations of diagonalizable groups split into character eigenspaces, The adjoint representation of an affine group scheme). This nonzero-weight definition uses no choice principle.

Assume the Axiom of Choice for the following supplemental structural facts and subgroup constructions (The Axiom of Choice). The reductive centralizer identity $C_G(T)=T$ and Lie fixed-point equality give ${\mathfrak{g}}_0=\mathrm{Lie}T=C_{\mathfrak{g}}(T)$ (Chevalley's centralizer theorem and reductive centralizers, The Lie functor: exactness, fixed points and generation). For $\alpha \in \mathrm{\Phi }$, put $T_{\alpha }=(\ker \alpha {)}_t=(\ker \alpha {)}_{\mathrm{r}\mathrm{e}\mathrm{d}}^{\circ }$, the maximal reduced subtorus of its kernel, of codimension one in $T$, and $G_{\alpha }=C_G(T_{\alpha })$. The root group is $U_{\alpha }=H_{(\alpha )}\subseteq G_{\alpha }$, attached to the semigroup of strictly positive rational multiples of $\alpha$ in $X(T)$; it is smooth connected unipotent and $T$-stable with Lie algebra ${⨁}_{\beta \in (\alpha )\cap \mathrm{\Phi }}{\mathfrak{g}}_{\beta }$. Its identity-concentrator construction takes place in $G_{\alpha }$, not in all of $G$ (Weight subgroups of a torus action, Cocharacter limit subgroups).

The Weyl group is $W(G,T)=N_G(T)\mathrm{/}T$. Under the stated AC premise it is a finite étale group scheme and acts faithfully on $X(T)$ (Milne21.1 and21.12). A Borel subgroup $B\supseteq T$ determines the positive roots ${\mathrm{\Phi }}^{+}(B)=\{\alpha \in \mathrm{\Phi }:{\mathfrak{g}}_{\alpha }\subseteq \mathrm{Lie}B\}$ and negative roots $-{\mathrm{\Phi }}^{+}(B)$ (Borel subgroups, maximal tori and Borel pairs, Character and cocharacter lattices of a split torus). Root groups are independent of the auxiliary cocharacter because $(\alpha )$ is intrinsic and its smooth connected subgroup is characterized by its specified Lie weight subspace. Two Borels containing $T$ give the same positive roots exactly when they are equal (Milne21.23 and21.35). These structural facts inherit the explicit AC premise; the definition of a root as a nonzero adjoint character above remains choice-free.