Let $G$ be a connected, reductive group, $T$ a maximal torus, and $B$ a Borel subgroup containing $T$. To each regular cocharacter $\lambda$ it is possible to assign a Borel subgroup $B(\lambda)$, in such a way that for an root $\alpha \in \Phi(G)$, we have $\alpha \in \Phi(B(\lambda))$ if and only if $\langle \alpha, \lambda \rangle > 0$.
Write $B = B(\lambda)$ for some $\lambda$, $U = B_u$, and for $w = nT$ in the Weyl group, let $U_w' = U \cap nU^-n^{-1}$, where $U^-$ is the unipotent part of $B(-\lambda)$.
As part of the proof of the Bruhat decomposition, one shows that$$f:U_w' \times B \rightarrow BnB, (x,y) \mapsto xny$$ is an isomorphism of varieties. I understand how to show that this mapping is bijective, but the last thing to do is to show that the map of tangent spaces $(df)_e$ is an isomorphism. I don't understand how to do this because, while I have a nice description of the tangent space of $U_w' \times B$ at $(1_G,1_G)$, I don't understand how to work with the tangent space of $BnB$ at $n$.
As it stands, $BnB$ is some algebraic variety, and its tangent space is defined abstractly. My reference is Borel, Linear Algebraic Groups, section 14.12.