I have been trying to do computations for objects of the based loop group and have been embarrassingly frustrated by the following:
Let $G$ be a compact, connected, simply connected Lie group with Lie algebra $\mathfrak g$. Let $\gamma: S^1 \to G$ be a based curve of class $H^1$ (or $C^\infty$ if you prefer) satisfying $\gamma(0)=e_G$. Now $\gamma'(\theta) \in T_{\gamma(\theta)} G$ for each $\theta \in S^1$. By standard abuse of notation, we may then think of $\gamma^{-1}(\theta)\gamma'(\theta)$ as a map $S^1 \to \mathfrak g$. A simple example might be the map $\gamma: S^1 \to SU(2)$ given by $$ \gamma(\theta) = \begin{pmatrix} e^{i\theta} & 0 \\ 0 & e^{-i\theta} \end{pmatrix}, \qquad \gamma'(\theta) = \begin{pmatrix} ie^{i\theta} & 0 \\ 0 & -ie^{-i\theta} \end{pmatrix},$$ for which $$ \gamma^{-1}(\theta) \gamma'(\theta) = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix} \in \mathfrak{su}(2),$$ is a constant map.
I have implicitly been thinking of $S^1$ in an additive coordinate and doing calculations as such, but it seems to me that I should equivalently be able to do calculations in a multiplicative coordinate $z=e^{i\theta}$. However, in such coordinates my above example becomes $$ \gamma(z) = \begin{pmatrix} z & 0 \\ 0 & z^{-1} \end{pmatrix}, \qquad \gamma'(z) = \begin{pmatrix} 1 & 0 \\ 0 & -z^{-2} \end{pmatrix}$$ yielding $$ \gamma^{-1}(\theta) \gamma'(\theta) =\begin{pmatrix} z^{-1} & 0 \\ 0 & -z^{-1} \end{pmatrix}$$ which is not an element of $\mathfrak{su}(2)$ for almost every value of $z$ (indeed, the diagonal terms must be purely imaginary so this only holds for $z=\pm i)$.
Everything seems to be legitimate up to chain-rule, change-of-coordinate, etc, except things aren't working. I am certain it must be some trivial over site on my part, but I cannot seem to pinpoint the error.