Background: Let $G$ be a complex reductive group, $T$ a maximal torus, $N$ the normalizer of $T$ in $G$, and $W = N/T$ the Weyl group. All in all, we have a group extension $$ 1 \to T \to N \to W \to 1 $$ of a finite group by a torus. Say $N$ "splits" if this extension splits, that is, if $W$ lifts to a subgroup of $N$.

It is well known that $N$ splits when $G=\text{GL}_n$: here $T$ is the diagonal matrices, $N$ is the group of monomial matrices, and a splitting is given by the permutation matrices. On the other hand, it is not true for every $G$ that the torus normalizer splits, the simplest example being $\text{SL}_n$ ($n\geq2$), where one cannot use the permutation matrices because the odd ones have determinant $-1$. It's a fun exercise to work out the other classical groups.

The group $N$ comes up all over in Lie theory and it was first studied, as far as I know, by Tits in a 1966 paper (zbmath). More recently, Moshe Adrian (zbmath) has completely answered the question of when $N$ splits, for $G$ almost simple. In particular:

Theorem: If $G$ is almost simple and exceptional then $N$ splits if and only if $G$ is of type $G_2$.

This result is already suggested (and maybe even directly implied) by work of Gal't in exceptional finite reductive groups (mathscinet, in Russian). I'm interested in the "exceptional" case of $G_2$, where $W$ is the dihedral group of order $12$.

Question: For the group $G_2$, is there a "simple" or "conceptual" explanation why the torus normalizer splits?

For example, $G_2$ is the automorphism group of the octonions $\mathbb O$ and one should be able to describe everything in these terms. Already the book of Springer and Veldkamp (zbmath) describes $T$ (Lemma 2.3.1) as automorphisms of $\mathbb O$ and my hope is that someone on this site who has drunk deep from the spring of the octonions can also describe $N$ and the splitting.

Alternatively, Tits's group for $G_2$ is of order $48$ and one should be able to recognize it as a semidirect product, though the computations involved intimidate me.

Background: Let $G$ be a complex reductive group, $T$ a maximal torus, $N$ the normalizer of $T$ in $G$, and $W = N/T$ the Weyl group. All in all, we have a group extension $$ 1 \to T \to N \to W \to 1 $$ of a finite group by a torus. Say $N$ "splits" if this extension splits, that is, if $W$ lifts to a subgroup of $N$.

It is well known that $N$ splits when $G=\text{GL}_n$: here $T$ is the diagonal matrices, $N$ is the group of monomial matrices, and a splitting is given by the permutation matrices. On the other hand, it is not true for every $G$ that the torus normalizer splits, the simplest example being $\text{SL}_n$ ($n\geq2$), where one cannot use the permutation matrices because the odd ones have determinant $-1$. It's a fun exercise to work out the other classical groups.

The group $N$ comes up all over in Lie theory and it was first studied, as far as I know, by Tits in a 1966 paper (zbmath). More recently, Moshe Adrian (zbmath) has completely answered the question of when $N$ splits, for $G$ almost simple. In particular:

Theorem: If $G$ is almost simple and exceptional then $N$ splits if and only if $G$ is of type $G_2$.

This result is already suggested (and maybe even directly implied) by work of Gal't in exceptional finite reductive groups (mathscinet, in Russian). I'm interested in the "exceptional" case of $G_2$, where $W$ is the dihedral group of order $12$.

Question: For the group $G_2$, is there a "simple" or "conceptual" explanation why the torus normalizer splits?

For example, $G_2$ is the automorphism group of the octonions $\mathbb O$ and one should be able to describe everything in these terms. Already the book of Springer and Veldkamp (zbmath) describes $T$ (Lemma 2.3.1) as automorphisms of $\mathbb O$ and my hope is that someone on this site who has drunk deep from the spring of the octonions can also describe $N$ and the splitting.

Alternatively, Tits's group for $G_2$ is of order $48$ and one should be able to recognize it as a semidirect product, though the computations involved intimidate me.

Background: Let $G$ be a complex reductive group, $T$ a maximal torus, $N$ the normalizer of $T$ in $G$, and $W = N/T$ the Weyl group. All in all, we have a group extension $$ 1 \to T \to N \to W \to 1 $$ of a finite group by a torus. Say $N$ "splits" if this extension splits, that is, if $W$ lifts to a subgroup of $N$.

It is well known that $N$ splits when $G=\text{GL}_n$: here $T$ is the diagonal matrices, $N$ is the group of monomial matrices, and a splitting is given by the permutation matrices. On the other hand, it is not true for every $G$ that the torus normalizer splits, the simplest example being $\text{SL}_n$ ($n\geq2$), where one cannot use the permutation matrices because the odd ones have determinant $-1$. It's a fun exercise to work out the other classical groups.

The group $N$ comes up all over in Lie theory and it was first studied, as far as I know, by Tits in a 1966 paper (zbmath). More recently, Moshe Adrian (zbmath) has completely answered the question of when $N$ splits, for $G$ almost simple. In particular:

Theorem: If $G$ is almost simple and exceptional then $N$ splits if and only if $G$ is of type $G_2$.

This result is already suggested (and maybe even directly implied) by work of Gal't in exceptional finite reductive groups (mathscinet, in Russian). I'm interested in the "exceptional" case of $G_2$.

Question: For the group $G_2$, is there a "simple" or "conceptual" explanation why the torus normalizer splits?

For example, $G_2$ is the automorphism group of the octonions $\mathbb O$ and one should be able to describe everything in these terms. Already the book of Springer and Veldkamp (zbmath) describes $T$ (Lemma 2.3.1) as automorphisms of $\mathbb O$ and my hope is that someone on this site who has drunk deep from the spring of the octonions can also describe $N$ and the splitting.

Alternatively, Tits's group for $G_2$ is of order $48$ and one should be able to recognize it as a semidirect product, though the computations involved intimidate me.

Background: Let $G$ be a complex reductive group, $T$ a maximal torus, $N$ the normalizer of $T$ in $G$, and $W = N/T$ the Weyl group. All in all, we have a group extension $$ 1 \to T \to N \to W \to 1 $$ of a finite group by a torus. Say $N$ "splits" if this extension splits, that is, if $W$ lifts to a subgroup of $N$.

It is well known that $N$ splits when $G=\text{GL}_n$: here $T$ is the diagonal matrices, $N$ is the group of monomial matrices, and a splitting is given by the permutation matrices. On the other hand, it is not true for every $G$ that the torus normalizer splits, the simplest example being $\text{SL}_n$ ($n\geq2$), where one cannot use the permutation matrices because the odd ones have determinant $-1$. It's a fun exercise to work out the other classical groups.

The group $N$ comes up all over in Lie theory and it was first studied, as far as I know, by Tits in a 1966 paper (zbmath). More recently, Moshe Adrian (zbmath) has completely answered the question of when $N$ splits, for $G$ almost simple. In particular:

Theorem: If $G$ is almost simple and exceptional then $N$ splits if and only if $G$ is of type $G_2$.

This result is already suggested (and maybe even directly implied) by work of Gal't in exceptional finite reductive groups (mathscinet, in Russian). I'm interested in the "exceptional" case of $G_2$, where $W$ is the dihedral group of order $12$.

Question: For the group $G_2$, is there a "simple" or "conceptual" explanation why the torus normalizer splits?

For example, $G_2$ is the automorphism group of the octonions $\mathbb O$ and one should be able to describe everything in these terms. Already the book of Springer and Veldkamp (zbmath) describes $T$ (Lemma 2.3.1) as automorphisms of $\mathbb O$ and my hope is that someone on this site who has drunk deep from the spring of the octonions can also describe $N$ and the splitting.

Alternatively, Tits's group for $G_2$ is of order $48$ and one should be able to recognize it as a semidirect product, though the computations involved intimidate me.