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Let $T$ be a split torus over a field $F$.

The Weyl group is by definition a subgroup of the group of outer morphisms (defined over $F$).

There are only two automorphism (the inverse and the identity) of $T$ (defined over F), so the group of outer automorphism is isomorphic to $\mathbb{Z}/2$.

Both can realized by the conugation via the non-trivial element $\begin{pmatrix} 0 & -1 \newline 1 & 0 \end{pmatrix}$ and by the identity elment respectively.

So the Weyl group has at least two elements, hence exactly two elements.

Similiar you can argue for the maximal split torus in $SL(n)$ or $GL(n)$.

For non-split tori, the question is more difficult to answer. I know only the $GL(n)$-situation. In $GL(2)$, the tori is isomorphic to a multiplicative group of a quadratic field extensions, and the outer automrophisms automorphisms of the tori are somorphic isomorphic to the Galois group. Now these can be realized by conjugation of elements $GL(2)$, but I am not sure whether they can not be realized by elements of determinant $1$.In fact, I would bet against it.

2 added 468 characters in body

Let $T$ be a split torus over a field $F$.

The Weyl group is by definition a subgroup of the group of outer morphisms (defined over $F$).

There are only two automorphism (the inverse and the identity) of $T$ (defined over F), so the group of outer automorphism is isomorphic to $\mathbb{Z}/2$.

The

Both can realized by the conugation via the non-trivial element $\begin{pmatrix} 0 & -1 \newline 1 & 0 \end{pmatrix}$ and by the identity elment respectively.

So the Weyl group has at least two elements, hence exactly two elements.

Similiar you can argue for the maximal split torus in $SL(n)$ or $GL(n)$.

For non-split tori, the question is more difficult to answer. I know only the $GL(n)$-situation. In $GL(2)$, the tori is isomorphic to a multiplicative group of a quadratic field extensions, and the outer automrophisms of the tori are somorphic to the Galois group. Now these can be realized by conjugation of elements $GL(2)$, but I am not sure whether they can be realized by elements of determinant $1$. In fact, I would bet against it.

1

Let $T$ be a split torus over a field $F$.

The Weyl group is by definition a subgroup of the group of outer morphisms (defined over $F$).

There are only two automorphism (the inverse and the identity) of $T$ (defined over F), so the group of outer automorphism is isomorphic to $\mathbb{Z}/2$.

The realized by the conugation via the non-trivial element $\begin{pmatrix} 0 & -1 \newline 1 & 0 \end{pmatrix}$ and by the identity.

So the Weyl group has at least two elements, hence exactly two elements.

Similiar you can argue for the maximal split torus in $SL(n)$ or $GL(n)$.