is now not possible. Instead we can (uniquely) write = + , \[const\] where ${\mbox{$| \phi_{BC} \rangle$}}$ is the vector of $R(\rho_{BC})$ which is orthogonal to ${\mbox{$| b_1 \rangle$}}{\mbox{$| c_1 \rangle$}}$, and ${\mbox{$| a_1 \rangle$}}$ and ${\mbox{$| a_2 \rangle$}}$ are given by ${\mbox{$\langle b_1|\langle c_1  | \psi \rangle$}}$ and ${\mbox{$\langle \phi_{BC}  | \psi \rangle$}}$. By means of LU (\[const\]) can be always rewritten as = (+ )&&\
+&&( + ). \[Wclass2\] Indeed, we first take ${\mbox{$| b_1 \rangle$}}{\mbox{$| c_1 \rangle$}}$ into ${\mbox{$| 0 \rangle$}}{\mbox{$| 0 \rangle$}}$. Then, since ${\mbox{$| \phi_{BC} \rangle$}}$ has been chosen orthogonal to ${\mbox{$| b_1 \rangle$}}{\mbox{$| c_1 \rangle$}}$, it must become $x{\mbox{$| 01 \rangle$}} +y{\mbox{$| 10 \rangle$}}+z {\mbox{$| 11 \rangle$}}$. By requiring that a linear combination of these two vectors has no second product vector we obtain that $z=0$ [@product]. In addittion the coefficients $ \sqrt{a}\equiv x,~\sqrt{b}\equiv
y,~\sqrt{c}$ and $\sqrt{d}$ can be made positive by absorbing the three relative phases into the definition of state ${\mbox{$| 1 \rangle$}}$ of subsystems $A$, $B$ and $C$. Thus case (i) has been taken into the form (\[Wclass2\]) by just using LU. Before we showed that 2 terms could not suffice in a product decomposition of the state. Now we see 