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The notions of a perfect element and an admissible element of the free modular lattice $D^r$ generated by $r\ge 1$ elements are introduced by Gelfand and Ponomarev. We recall that an element $a\in D$ of a modular lattice $L$ is said to be {\it perfect}, if, for each finite dimension indecomposable $K$-linear representation ${\rho }_X:L\to \mathcal{L}(X)$ over any field $K$, the image ${\rho }_X(a)\subseteq X$ of $a$ is either zero, or ${\rho }_X(a)=X$, where $\mathcal{L}(X)$ is the lattice of all vector $K$-subspaces of $X$.

Gelfand and Ponomarev gave a complete classification of such elements in the lattice $D^4$, associated to the extended Dynkin diagram ${\tilde{\mathbb{D}}}_4$, and also in $D^r$, where $r>4$.

The main aim of this paper is to classify all the {\it admissible elements} and all the perfect elements in the Dedekind lattice $D^{2,2,2}$ generated by six elements that is associate to the extended Dynkin diagram ${\tilde{\mathbb{E}}}_6$. Gelfand and Ponomarev constructed admissible elements of the lattice $D^r$ recurrently whereas we suggest a direct method for creatin admissible elements. Using this method we also construct admissible elements for $D^4$ and show that these elements coincide, modulo linear equivalence, with admissible elements constructed by Gelfand and Ponomarev. Admissible sequences and admissible elements for $D^{2,2,2}$ (resp. $D^4$) form $14$ classes (resp. $8$ classes) and possess a certain periodicity.

Our classification of perfect elements for $D^{2,2,2}$ is based on the description of admissible elements. The constructed set $H^{+}$ of perfect elements is the union of $64$-element distributive lattices $H^{+}(n)$, and $H^{+}$ is the distributive lattice itself. The lattice of perfect elements $B^{+}$ obtained by Gelfand and Ponomarev for $D^4$ can be imbedded into the lattice of perfect elements $H^{+}$, associated with $D^{2,2,2}$.

Herrmann constructed perfect elements $s_n$, $t_n$, $p_{i,n}$ in $D^4$ by means of certain endomorphisms ${\gamma }_{ij}$ and showed that these perfect elements coincide with the Gelfand-Ponomarev perfect elements modulo linear equivalence. We show that the admissible elements in $D^4$ are also obtained by means of Herrmann's endomorphisms ${\gamma }_{ij}$. Herrmann's endomorphism ${\gamma }_{ij}$ and the {\it elementary map} of Gelfand-Ponomarev ${\varphi }_i$ act, in a sense, in opposite directions, namely the endomorphism ${\gamma }_{ij}$ adds the index to the beginning of the admissible sequence, and the elementary map ${\varphi }_i$ adds the index to the end of the admissible sequence.