Let $G$ be a connected split reductive group over a field $k$ of characteristic $0$. Let $T$ and $T'$ be two split maximal tori of $G$ and $B \supset T, B' \supset T'$ be two Borel subgroups of $G$.

Let $W(G,T)=N_G(T)/T$ and $W(G,T')=N_G(T')/T'$ be the Weyl groups of $G$ relative to $T$ and $T'$ respectively.

Let $w \in W(G,T)$ be the relative position of $B$ and $B'$ i.e. there exists $b \in B$ such that $bB'b^{-1} = wBw^{-1}$ (where I see $w$ as an element of $G$).

My question is : is there an explicit description of the relative position of $B$ and $B'$ as an element of $W(G,T')$ ?

Note that given any element $g$ of $G$ such that $gTg^{-1} = T'$ then the map $\gamma \mapsto g\gamma g^{-1}$ induces a bijection between $W(G,T)$ and $W(G,T')$. Ideally I would like to find a concrete recipe for finding a $g$ such that the relative position of $B$ and $B'$ as an element of $W(G,T')$ is $gwg^{-1}$.