Let $P$ be a three-holed sphere with its usual presentation $$ \pi_1(P) = \langle U, V, W | UVW=1 \rangle. $$
A hyperbolic structure on $P$ is given by a Fuchsian representation $\rho:\pi_1(P) \to \operatorname{PSL}(2, \mathbb{R})$.
That is $\rho(U), \rho(V)$ and $\rho(W)$ are now represented by matrices subject to the condition that $\rho(U) \rho(V)= \rho(W)^{-1}$ imposed by the presentation of $\pi_1(P)$.
I am reading a paper in which the author finds that $$\rho(U) = \begin{pmatrix} e^{a/2} & \kappa \\ 0 & e^{-a/2} \end{pmatrix}$$ and $$ \rho(V) = \begin{pmatrix} e^{-b/2} & 0 \\ 1 & e^{b/2} \end{pmatrix}$$ where $a$ and $b$ are the respective lengths of the boundaries $U$ and $V$.
I know that since $\rho(U)$, $\rho(V)$ and $\rho(W)$ are hyperbolic elements of $\operatorname{PSL}(2, \mathbb{R})$, there exists a $\operatorname{PSL}(2, \mathbb{R})$-conjugation such that $\rho(U) = \begin{pmatrix} e^{a/2} & 0 \\ 0 & e^{-a/2} \end{pmatrix} $ and similarly for $\rho(V)$.
However, how were the above matrix representations including the terms $\kappa$ (resp. $1$) obtained?