namely = + . \[2deco\] Then, also according to the previous lemma, ${\mbox{$| b_1 \rangle$}}{\mbox{$| c_1 \rangle$}}$ and ${\mbox{$| b_2 \rangle$}}{\mbox{$| c_2 \rangle$}}$ have to span the range of $\rho_{BC}$, R$(\rho_{BC})$.

But R$(\rho_{BC})$ is a two dimensional subspace of $\C^2\otimes\C^2$. Therefore it always contains either only one or only two product states [@STV] \[unless R$(\rho_{BC})$ was supported in $\C\otimes\C^2$ or $\C^2\otimes\C$, but we already excluded this possibility because we are considering r$(\rho_B)=$r$(\rho_C)=2$\]. Notice that a two-term decomposition (\[2deco\]) requires that R$(\rho_{BC})$ contains at least two product vectors. Only one product vector in R$(\rho_{BC})$, and thus the impossibility of decomposition (\[2deco\]), is going to be precisely the trait of the states in the W-class.

[**GHZ-class**]{}\
Let us suppose first that R$(\rho_{BC})$ contains two product vectors, ${\mbox{$| b_1 \rangle$}}{\mbox{$| c_1 \rangle$}}$ and ${\mbox{$| b_2 \rangle$}}{\mbox{$| c_2 \rangle$}}$. Then decomposition (\[2deco\]) is possible, and actually unique, with ${\mbox{$| a_i \rangle$}}= {\mbox{$\langle \xi_i  | \psi \rangle$}}$, $i=1,2$, where ${\mbox{$| \xi_i \rangle$}}$ are the two vectors supported in $R(\rho_{BC})$ that are biorthonormal to the ${\mbox{$| b_i \rangle$}}{\mbox{$| c_i \rangle$}}$. In this case we can use LU in order to take ${\mbox{$| \psi \rangle$}}$ into the useful standard product form (see also [@Ac00]) |\_[GHZ]{}=(c\_|0|0|0+s\_e\^[i]{}|\_A|\_B|\_C), \[GHZclass\] where 