The problem of representation of an algebraic lattice $L$ by the lattice $C_W(\mathcal A)$ of weak congruences of an algebra $\mathcal A$ (i.e. of all congruences on all the subalgebras of $A$) is closely related to the localization of a diagonal relation $\Delta$ in $C_W(\mathcal A)$. $\Delta$ is a co-distributive element in $C_W(\mathcal A)$ [6], and it can also be neutral [6] and exceptional (as shown here). Here we shall discuss the following question: If a is a co-distributive (neutral, exceptional) element of an algebraic lattice $L$, is there an algebra $\mathcal A$, such that $C_W(\mathcal A)\simeq L$, and that $f(a)=\Delta$ under that isomorphism?