  For linear differential equations of the form $u'(t)=[A + B(t)] u(t)$,
$t\geq0$, with a possibly unbounded operator $A$, we construct and deduce error
bounds for two families of second-order exponential splittings. The role of
quadratures when integrating the twice-iterated Duhamel's formula is
reformulated: we show that their choice defines the structure of the splitting.
Furthermore, the reformulation allows us to consider quadratures based on the
Birkhoff interpolation to obtain not only pure-stages splittings but also those
containing derivatives of $B(t)$ and commutators of $A$ and $B(t)$. In this
approach, the construction and error analysis of the splittings are carried out
simultaneously. We discuss the accuracy of the members of the families.
Numerical experiments are presented to complement the theoretical
consideration.
