Conditional density of degenerate multivariate normal

Question

Let $X_{12},X_{13},X_{14},X_{23},X_{24},X_{34}$ be identically normal $N(\mu,\sigma^2)$ such that every linear combination among $X_{ij}$'s is also normal, $corr(X_{ij},X_{rs})=\rho$ if $|\{i,j\}\cap\{r,s\}|=1$ and $0$ otherwise. Define $Z=(X_{12},X_{13},X_{14},X_{23},X_{24},X_{34})^T$ and $Y=(X_{12},X_{23},X_{34},X_{14})^T.$ I wish to know the conditional density of $Z|Y.$ Note that since every linear combination of $X_{ij}$'s is normal then $Z$ and $Y$ are also normal. This implies $$f_{Z|Y}(u,v)=f_Z(a,b,c,d,u,v)/f_Y(a,b,c,d).$$ However, a straightforward computation gives $Cov(Z)$ is singular. So, How would I do that? Any advice?