  This article discusses the twisted adjoint action
$\mathrm{Ad}_{g}^{\kappa}:G\rightarrow G$, $x\mapsto gx\kappa(g^{-1})$ given by
a Dynkin diagram automorphism $\kappa\in\mathrm{Aut}(G)$, where $G$ is compact,
connected, simply connected and simple. The first aim is to recover the
classification of $\kappa$-twisted conjugacy classes by elementary means,
without invoking the non-connected group $G\rtimes\langle\kappa\rangle$. The
second objective is to highlight several properties of the so-called
\textit{twining characters} $\tilde{\chi}^{(\kappa)}:G\rightarrow\mathbb{C}$,
as defined by Fuchs, Schellekens and Schweigert. These class functions
generalize the usual characters, and define $\kappa$-twisted versions
$\tilde{R}^{(\kappa)}(G)$ and $\tilde{R}_{k}^{(\kappa)}(G)$
($k\in\mathbb{Z}_{>0}$) of the representation and fusion rings associated to
$G$. In particular, the latter are shown to be isomorphic to the representation
and fusion rings of the \textit{orbit Lie group} $G_{(\kappa)}$, a simply
connected group obtained from $\kappa$ and the root data of $G$.
