  When considering the unit group of $\mathcal{O}_F G$ ($\mathcal{O}_F$ the
ring of integers of an abelian number field $F$ and a finite group $G$) certain
components in the Wedderburn decomposition of $FG$ cause problems for known
generic constructions of units; these components are called exceptional.
Exceptional components are divided into two types: type 1 are division rings,
type 2 are $2 \times 2$-matrix rings. For exceptional components of type 1 we
provide infinite classes of division rings by describing the seven cases of
minimal groups (w.r.t. quotients) having those division rings in their
Wedderburn decomposition over $F$. We also classify the exceptional components
of type 2 appearing in group algebras of a finite group over number fields $F$
by describing all 58 finite groups $G$ having a faithful exceptional Wedderburn
component of this type in $FG$.
