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Now, at last, the mythic (and canonical!) "field of definition" $F_a$ in the question (which will be finite separable over $F$) emerges from the shadows. This rests on the following fundamental result (the proof of which is part of the same circle of ideas that prove the Theorem stated in my answer to Automorphism of restriction of scalars):

Now, at last, the mythic (and canonical!) "field of definition" $F_a$ in the question (which will be finite separable over $F$) emerges from the shadows. This rests on the following fundamental result (the proof of which is part of the same circle of ideas that prove the Theorem stated in my answer to Automorphism of restriction of scalars):

Theorem. If $G$ is a nontrivial connected semisimple group over a field $F$ and it is simply connected then $G \simeq {\rm{R}}_{F'/F}(G')$ for a nonzero finite etale $F$-algebra and a smooth affine $F'$-group $G'$ whose fiber over each factor field of $F'$ is connected semisimple, simply connected, and absolutely simple. Moreover, the pair $(F'/F, G')$ is unique up to unique isomorphism in the sense that if $(F''/F, G'')$ is a second such pair then any $F$-isomorphism ${\rm{R}}_{F''/F}(G'') \simeq {\rm{R}}_{F'/F}(G')$ arises from a unique pair $(\alpha, \varphi)$ consisting of an $F$-algebra isomorphism $\alpha:F' \simeq F''$ and a group isomorphism $\varphi:G'\simeq G''$ over $\alpha$.

The maximal $k$-tori in the Borel subgroups are the Weil restriction of ${\rm{GL}}_1$ from the finite etale $k$-algebra $k'$ associated to the ${\rm{Gal}}(k_s/k)$-action on $\Delta$: we associated to each Galois-orbit of vertices the subfield of $k_s$ associated to the open subgroup fixing a choice of vertex in the orbit. In particular, if there are at least 2 Galois orbits then the $k$-rank of the associated quasi-split form is at least 2 since ${\rm{R}}_{K/k}({\rm{GL}}_1)$ contains ${\rm{GL}}_1$ as a $k$-subgroup for any finite separable extension field $K/k$. But we assumed $H$ has $k$-rank equal to 1, so we only need to consider cases when ${\rm{Gal}}(k_s/k)$ acts transitively on the set of vertices of $\Delta$. This can only happen when ${\rm{Aut}}(\Delta)$ acts transitively on $\Delta$. Inspection of the diagrams for A$_n$ ($n \ge 2$), D$_n$ ($n \ge 4$), and E$_6$ shows that the latter only happens for the A$_2$ diagram. The quasi-split groups of type A with absolute rank $n \ge 2$ are exactly the unitary groups ${\rm{SU}}_n(K/k)$ associated to quadratic Galois extensions $K/k$ (which have $k$-rank equal to $n-1$). The extension $K/k$ is unique up to isomorphism since it is the splitting field for the Galois action on the absolute diagram.

Theorem. If $G$ is a nontrivial connected semisimple group over a field $F$ and it is simply connected then $G \simeq {\rm{R}}_{F'/F}(G')$ for a nonzero finite etale $F$-algebra $F'$ and a smooth affine $F'$-group $G'$ whose fiber over each factor field of $F'$ is connected semisimple, simply connected, and absolutely simple. Moreover, the pair $(F'/F, G')$ is unique up to unique isomorphism in the sense that if $(F''/F, G'')$ is a second such pair then any $F$-isomorphism ${\rm{R}}_{F''/F}(G'') \simeq {\rm{R}}_{F'/F}(G')$ arises from a unique pair $(\alpha, \varphi)$ consisting of an $F$-algebra isomorphism $\alpha:F' \simeq F''$ and a group isomorphism $\varphi:G'\simeq G''$ over $\alpha$.

The maximal $k$-tori in the Borel subgroups are the Weil restriction of ${\rm{GL}}_1$ from the finite etale $k$-algebra $k'$ associated to the ${\rm{Gal}}(k_s/k)$-action on $\Delta$: we associated to each Galois-orbit of vertices the subfield of $k_s$ associated to the open subgroup fixing a choice of vertex in the orbit. In particular, if there are at least 2 Galois orbits then the $k$-rank of the associated quasi-split form is at least 2 since ${\rm{R}}_{K/k}({\rm{GL}}_1)$ contains ${\rm{GL}}_1$ as a $k$-subgroup for any finite separable extension field $K/k$. But we assumed $H$ has $k$-rank equal to 1, so we only need to consider cases when ${\rm{Gal}}(k_s/k)$ acts transitively on the set of vertices of $\Delta$. This can only happen when ${\rm{Aut}}(\Delta)$ acts transitively on $\Delta$. Inspection of the diagrams for A$_n$ ($n \ge 2$), D$_n$ ($n \ge 4$), and E$_6$ shows that the latter only happens for the A$_2$ diagram. The quasi-split groups of type A with absolute rank $n \ge 2$ are exactly the special unitary groups ${\rm{SU}}_{n+1}(K/k)$ associated to quadratic Galois extensions $K/k$ (and these $k$-groups have $k$-rank equal to $n-1$). The extension $K/k$ is unique up to isomorphism since it is the splitting field for the Galois action on the absolute diagram.

This is explained by the Borel-Tits relative structure theory for connected reductive groups over arbitrary fields. In particular, there is no need whatsoever to assume $F$ is perfect (recall that all global fields and all non-archimedean local fields should be treated on equal footing whenever possible, and the global and local function fields are never perfect). The explanation seems long when written out, but the underlying principles are rather natural and clean.

  I will use Roman letters to denote roots, so "$a$" rather than "$\alpha$". I should begin by correcting what seems to be a misconception: the correct definition of $F_a$ shows that it is generally not a Galois extension of $F$. Rather, it is a finite separable extension of $F$ (canonical when $a$ is non-multipliable in the relative root system, but seemingly only quadratic Galois over something canonical in the multipliable case -- if anyone can see a way to canonify these latter cases then please let me know!), and the Galois orbit of roots in question is not at all indexed by the Galois group for $F_a$ over $F$ (which won't make sense when $F_a/F$ is not Galois) but rather by the set of $F$-embeddings of $F_a$ into a fixed separable closure $F_s$ (sanity check: ${\rm{Gal}}(F_s/F)$ does act naturally and transitively on that set, regardless of whether or not $F_a/F$ is Galois!).

We are going to postpone the quasi-split hypothesis for as long as possible, to identify most clearly where exactly this assumption is truly essential. But at the risk of ruining all suspense, it should be said that we will prove that for a relative root $a$ in a chosen basis for the relative root system (these correspond exactly to a Galois orbit in a basis of the absolute root system consisting of absolute roots with nontrivial restriction to a maximal split torus), the $a$-root space that may have massive dimension as an $F$-vector space is a line over a finite separable extension $F_a$ of $F$. The construction of this enhanced linear structure over such an extension will make clear the sense in which it is canonical for non-multipliable $a$ (and nearly so for multipliable $a$), and will use in a crucial way the quasi-split hypothesis.

For an arbitrary connected reductive $F$-group $G$ (not yet assumed to be quasi-split, to clarify ideas), let $S$ be a maximal split $F$-torus. The Borel-Tits relative structure theory over arbitrary fields ensures that:

Proposition. If $H$ is a connected semisimple group over a field $k$ and it is absolutely simple, simply connected, and quasi-split with $k$-rank equal to $1$ then $H \simeq {\rm{SL}}_2$ or $H \simeq {\rm{SU}}_3(K/k)$ is the quasi-split unitary group associated to a quadratic Galois extension $K/k$ that is unique up to $k$-isomorphism.

This is explained by the Borel-Tits relative structure theory for connected reductive groups over arbitrary fields. In particular, there is no need to assume $F$ is perfect. The explanation below is long when written out, but the underlying principles are rather natural and clean. Moreover, we will find a collection of root vectors satisfying the stronger requested Galois-equivariance property as near the start of the question. I will use Roman letters to denote roots, so "$a$" rather than "$\alpha$".

Consider a relative root $a$ in a chosen basis for the relative root system (it corresponds to a Galois orbit in a basis of the absolute root system such that the members of the orbit have nontrivial restriction to a maximal split torus, as we review below). The $a$-root space may have massive dimension as an $F$-vector space, but we will show that it is naturally a line over a finite separable extension $F_a$ of $F$. The construction of this enhanced linear structure over such an extension will make clear the sense in which $F_a/F$ and that $F_a$-linear structure refining the given $F$-linear structure is canonical for non-multipliable $a$ (and nearly canonical for multipliable $a$). This will use in a crucial way the quasi-split hypothesis.

First, we fix what seems to be a misconception: the correct definition of $F_a$ will show that it is generally not a Galois extension of $F$. Rather, it is a finite separable extension of $F$, and the Galois orbit of absolute roots in question is not naturally indexed by the Galois group for $F_a$ over $F$ (there is no natural base point for the orbit, and no such Galois group when $F_a/F$ is not Galois) but rather is naturally indexed by the set of $F$-embeddings of $F_a$ into a fixed separable closure $F_s$ (sanity check: ${\rm{Gal}}(F_s/F)$ does act naturally and transitively on that set, regardless of whether or not $F_a/F$ is Galois!).

We are going to postpone the quasi-split hypothesis for as long as possible, to identify most clearly where exactly this assumption is truly essential. For an arbitrary connected reductive $F$-group $G$ (not yet assumed to be quasi-split, to clarify ideas), let $S$ be a maximal split $F$-torus. The Borel-Tits relative structure theory over arbitrary fields ensures that:

Proposition. If $H$ is a connected semisimple group over a field $k$ and it is absolutely simple, simply connected, and quasi-split with $k$-rank equal to $1$ then $H$ is either $k$-isomorphic to ${\rm{SL}}_2$ or $H$ is the quasi-split special unitary group ${\rm{SU}}_3(K/k)$ associated to a quadratic Galois extension $K/k$. In the latter case, $K/k$ is determined by $G$ up to $k$-isomorphism.