  Let $G$ be a connected semi-simple algebraic group of adjoint type over an
algebraically closed field, and let $\overline{G}$ be the wonderful
compactification of $G$. For a fixed pair $(B, B^-)$ of opposite Borel
subgroups of $G$, we look at intersections of Lusztig's $G$-stable pieces and
the $B^-\times B$-orbits in $\overline{G}$, as well as intersections of $B
\times B$-orbits and $B^- \times B^-$-orbits in $\overline{G}$. We give
explicit conditions for such intersections to be non-empty, and in each case,
we show that every non-empty intersection is smooth and irreducible, that the
closure of the intersection is equal to the intersection of the closures, and
that the non-empty intersections form a strongly admissible partition of
$\overline{G}$.
