  It is well-known that every polyhedral cone is finitely generated (i.e.
polytopal), and vice versa. Surprisingly, the two notions differ almost always
for non-commutative versions of such cones. This was obtained as a byproduct in
an earlier paper. In this note we give a direct and constructive proof of the
statement. Our proof also yields a surprising quantitative result: the
difference of the two notions can always be seen at the first level of
non-commutativity, i.e. for matrices of size $2$, independent of dimension and
complexity of the initial convex cone.
