Given a semisimple Lie group G, let T be a maximal torus, W be the Weyl group defined as the quotient N(T)/C(T), where N(T) denotes the normalizer of T and C(T) denotes the centralizer. 


Two questions are: 


1. How many ways are there we can realize W as a subset of G?
2. Can we always realize W as a subgroup of G?



Can we realize Weyl group as a subgroup?
In general it is not possible to embed the Weyl group $W$ in the group $G$: already you can see this for $SL\_2(\mathbb C)$, where the Weyl group has order $2$: if the torus fixes the lines spanned by $e\_1$ and $e\_2$ respectively, you want to pick the linear map taking $e\_1$ to $e\_2$ and $e\_2$ to $e\_1$, but this has determinant $-1$. A lift of $W$ to $N(T)$ must be an element of order $4$ not $2$, say $e\_1 \mapsto -e\_2$ and $e\_2 \mapsto e\_1$. 


In fact, Tits has shown ([Normalisateurs de Tores I. Groupes de Coxeter Étendus (Journ. Alg. 4, 1966, pp. 96-116](https://doi.org/10.1016/0021-8693(66)90053-6)) that this is essentially the only obstruction: the Weyl group can always be lifted to a group $\tilde{W}$ inside $G$ which is an extension of $W$ by an elementary abelian $2$-group of order $2^l$ where $l$ is the number of simple roots. If I recall correctly, this lift is then unique up to conjugation.


