a geodesic curve. Write $v=v^1\sigma_1+v^2\sigma_2+v^3\sigma_3$ and compute $S(t)\cdot \sigma_i$ to find $$v(t) = v^1\sigma_1 + v^2\sigma_2 + v^3\begin{pmatrix}
    e^{at} & 0\\
    0 & -e^{-at}
    \end{pmatrix}.$$ Compute in ${\mathbb{R}}^{1,3}$ $$\frac{Dv(t)}{dt} = \left(\frac{dv(t)}{dt}\right)^T = av^3\left(\gamma(t)\right)^T = 0$$ to see that $v(t)$ is the parallel transport of $v$ along the geodesic $\gamma$.

If $S$ is not diagonal, write $S = QDQ^{-1}$ where $Q\in{\mathrm{SU}}(2)$ and $D\in{\mathcal{H}}_2^{++}\cap {\mathrm{SL}}(2,{\mathbb{C}})$ is diagonal. Then, $$S\cdot v = Q\cdot \left( D\cdot \left( Q^{-1}\cdot v \right) \right) = Q\cdot \Gamma_{{\mathrm{I}}_2}^{D\cdot {\mathrm{I}}_2}(Q^{-1}\cdot v).$$ But for all $A\in{\mathrm{SL}}(2,{\mathbb{C}})$, $p,q\in{\mathbb{H}}^3$ and $v\in T_p{\mathbb{H}}^3$, $$A\cdot \Gamma_p^qv = \Gamma_{A\cdot p}^{A\cdot q}A\cdot v$$ and thus $$S\cdot v = \Gamma_{{\mathrm{I}}_2}^pv.$$

The DPW method for CMC $H>1$ surfaces in ${\mathbb{H}}^3$ {#sectionDPWMethod}
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#### Loop groups. {#sectionLoopGroups}

In the DPW method, a whole family of surfaces is constructed, depending on a spectral parameter ${\lambda}$. This parameter will always be in one of the following subsets of ${\mathbb{C}}$ ($\rho>1$): $${\mathbb{S}}^1 = \left\{ {\lambda}\in{\mathbb{C}}\mid|{\lambda}|=1 \right\}, \quad {\mathbb{A}}_\rho = \left\{ {\lambda}\in{\mathbb{C}}\mid \rho^{-1}<|{\lambda}|<\rho \right\}, \quad {\mathbb{D}}_\rho = \left\{ {\lambda}\in{\mathbb{C}}\mid |{\lambda}|<\rho \right\}.$$ Any smooth map $f:{\mathbb{S}}^1\longrightarrow {\mathcal{M}}(2,{\mathbb{C}})$ can be decomposed into its Fourier series $$f({\lambda}) = \sum_{i\in{\mathbb{Z}}}f_i{\lambda}^i.$$ Let $|\cdot|$ denote a norm on ${\mathcal{M}}(2,{\mathbb{C}})$. Fix some $\rho>1$ and consider $${\left\Vertf\right\Vert}_\rho := \sum_{i\in{\mathbb{Z}}} |f_i|\rho^{|i|}.$$ Let 