It's a bit unclear to me precisely what you want, but the usual solution is take the abstract Cartan (discussed a bit here), which is defined to be $\mathbb{T}=B/[B,B]$ for any Borel $B$. This is canonical, since any two Borels are conjugate, and every Borel is self-normalizing so $\mathbb{T}$ is unique up to the map induced by conjugation by an element of $B$. The action of the Weyl group is given by looking at all the isomorphisms induced between $\mathbb{T}$ and any fixed torus $T$ by all the different Borels containing $T$ (which are all conjugate under $N(T)$). Composing one of these isomorphisms with the inverse of another gives all the elements of the Weyl group.

It's a bit unclear to me precisely what you want, but the usual solution is take the abstract Cartan (discussed a bit here), which is defined to be $\mathbb{T}=B/[B,B]$ for any Borel $B$. This is canonical, since any two Borels are conjugate, and every Borel is self-normalizing so $\mathbb{T}$ is unique up to the map induced by conjugation by an element of $B$. The action of the Weyl group is given by looking at all the isomorphisms induced between $\mathbb{T}$ and any fixed torus $T$ by all the different Borels containing $T$ (which are all conjugate under $N(T)$). Composing one of these isomorphisms with the inverse of another gives all the elements of the Weyl group.