  We construct for the first time conditionally decomposable $d$-polytopes for
$d \ge 4$. These examples have any number of vertices from $4d-4$ upwards. A
polytope is said to be conditionally decomposable if one polytope
combinatorially equivalent to it is decomposable (with respect to the Minkowski
sum) and another one combinatorially equivalent to it is indecomposable. In our
examples, one has $4d-2$ vertices and is the sum of a line segment and a
bi-pyramid over a prism. The other ones have $4d-4$ vertices, and one of them
is the sum of a line segment and a $2$-fold pyramid over a prism. We show that
the latter examples have the minimum number of vertices among all conditionally
decomposable $d$-polytopes that have a line segment for a summand.
