  Vasiliev generating system of higher-spin equations allowing to reconstruct
nonlinear vertices of field equations for higher-spin gauge fields contains a
free complex parameter $\eta$. Solving the generating system order by order one
obtains physical vertices proportional to various powers of $\eta$ and
$\bar{\eta}$. Recently $\eta^2$ and $\bar{\eta}^2$ vertices in the zero-form
sector were presented in 2009.02811 in the $Z$-dominated form implying their
spin-locality by virtue of $Z$-dominance Lemma of 1805.11941. However the
vertex of 2009.02811 had the form of a sum of spin-local terms dependent on the
auxiliary spinor variable $Z$ in the theory modulo so-called $Z$-dominated
terms, providing a sort of existence theorem rather than explicit form of the
vertex. The aim of this paper is to elaborate an approach allowing to
systematically account for the effect of $Z$-dominated terms on the final
$Z$-independent form of the vertex needed for any practical analysis. Namely,
in this paper we obtain explicit $Z$-independent spin-local form for the vertex
$\Upsilon^{\eta\eta}_{\omega CCC}$ for its $\omega CCC$-ordered part where
$\omega$ and $C$ denote gauge one-form and field strength zero-form higher-spin
fields valued in an arbitrary associative algebra in which case the order of
product factors in the vertex matters. The developed formalism is based on the
Generalized Triangle identity derived in the paper and is applicable to all
other orderings of the fields in the vertex.
