  We make two contributions to the study of polite combination in
satisfiability modulo theories.
  The first contribution is a separation between politeness and strong
politeness, by presenting a polite theory that is not strongly polite. This
result shows that proving strong politeness
  (which is often harder than proving politeness) is sometimes needed in order
to use polite combination.
  The second contribution is an optimization to the polite combination method,
obtained by borrowing from the Nelson-Oppen method. In its non-deterministic
form, the Nelson-Oppen method is based on guessing arrangements over shared
variables. In contrast, polite combination requires an arrangement over
\emph{all} variables of the shared sort (not just the shared variables). We
show that when using polite combination, if the other theory is stably infinite
with respect to a shared sort, only the shared variables of that sort need be
considered in arrangements, as in the Nelson-Oppen method.
  Reasoning about arrangements of variables is exponential in the worst case,
so reducing the number of variables that are considered has the potential to
improve performance significantly.
  We show preliminary evidence for this in practice by demonstrating a speed-up
on a smart contract verification benchmark.
