TITLE: About the intrinsic definition of the Weyl group of complex semisimple Lie algebras
QUESTION [15 upvotes]: It may be a easy question for experts. 
The definition of the Weyl group of a complex semisimple Lie algebra $\mathfrak{g}$ is well-known: We first $\textbf{choose}$ a Cartan subalgebra $\mathfrak{h}$ and we have the root space decomposition. The Weyl group now is the group generated by the reflections according to roots.
Naively this definition depends on the choices of the Cartan subalgebra $\mathfrak{h}$. Of course we can prove that for different choices the resulting Weyl groups are isomorphic.
My question is: can we define the Weyl group intrinsically such that we don't need the do check the unambiguity. 
One thought is: we have the abstract Cartan subalgebra $\mathfrak{H}:=\mathfrak{b}/[\mathfrak{b},\mathfrak{b}]$ of $\mathfrak{g}$ (which is in fact not a subalgebra of $\mathfrak{g}$). Can we define the Weyl group along this way? Again is there any references for this?

REPLY [14 votes]: Sometimes when you define a group using an arbitrary choice of object and then show the choice of object doesn't matter, you could have defined a groupoid without making an arbitrary choice.  
For example, to define the fundamental group $\pi_1(X,x)$ of a path-connected space $X$ we need to choose a basepoint $x \in X$, but then we can show we get isomorphic groups no matter what basepoint we choose, with an isomorphism given by a homotopy class of paths between the basepoints.  To avoid this maneuver we can work with the fundamental groupoid of $X$, whose objects are points of $X$ and whose morphisms are homotopy classes of paths.  If $X$ is path-connected all objects in this groupoid are isomorphic, and thus the automorphism groups of all objects are isomorphic. The automorphism group of $x$ is just $\pi_1(X,x)$.  The fundamental groupoid is thus equivalent, as a category, to the one-object groupoid corresponding to the group $\pi_1(X,x)$.   But the advantage of the fundamental groupoid is that we can define it without choosing a basepoint, and it makes sense and works well even when $X$ is not path-connected.
Similarly, I think we can define the Weyl groupoid of a compact semisimple Lie group $G$ in a way that gives a groupoid equivalent to the usual Weyl group, but doesn't require a choice of maximal torus.  The idea should go like this.  The objects of the Weyl groupoid are maximal tori.  A morphism $f : T \to T'$ in the Weyl groupoid is a Lie group isomorphism of the form
$$   t \mapsto g t g^{-1}   \textrm{ for all } t \in T $$
for some $g \in G$.   If I did this right, the automorphism group of any object $T$ in the Weyl groupoid is the usual Weyl group 
$$  W_G(T) = N_G(T) / T ,$$
that is, the normalizer of $T \subset G$ modulo the centralizer of $T \subset G$, which is $T$ itself.  If this is true, the Weyl groupoid will be equivalent, as a groupoid, to the usual Weyl group $W_G(T)$ for any maximal torus $T$.