  We single out a class of states possessing only threetangle but distributed
all over four qubits. This is a three-site analogue of states from the
$W$-class, which only possess globally distributed pairwise entanglement as
measured by the concurrence. We perform an analysis for four qubits, showing
that such a state indeed exists. To this end we analyze specific states of four
qubits that are not convexly balanced as for $SL$ invariant families of
entanglement, but only affinely balanced. For these states all possible
$SL$-invariants vanish, hence they are part of the $SL$ null-cone. Instead,
they will possess at least a certain unitary invariant.
  As an interesting byproduct it is demonstrated that the exact convex roof is
reached in the rank-two case of a homogeneous polynomial $SL$-invariant measure
of entanglement of degree $2m$, if there is a state which corresponds to a
maximally $m$-fold degenerate solution in the zero-polytope that can be
combined with the convexified minimal characteristic curve to give a
decomposition of $\rho$. If more than one such state does exist in the zero
polytope, a minimization must be performed. A better lower bound than the
lowest convexified characteristic curve is obtained if no decomposition of
$\rho$ is obtained in this way.
