  We obtain a connection between admissible elements for quadruples and
Herrmann's endomorphisms. Herrmann constructed perfect elements $s_n$, $t_n$,
$p_{i,n}$ in $D^4$ by means of some endomorphisms 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 $\gamma_{ij}$. Endomorphism $\gamma_{ij}$ and
the elementary map of Gelfand-Ponomarev $\phi_i$ act, in a sense, in opposite
directions, namely the endomorphism $\gamma_{ij}$ adds the index to the start
of the admissible sequence, and the elementary map $\phi_i$ adds the index to
the end of the admissible sequence.
