Manifest form of the spin-local higher-spin vertex ${{\rm Y}}_{\omega CCC}^{\eta \eta }$

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 $\overline{\eta }$. Recently ${\eta }^2$ and ${\overline{\eta }}^2$ vertices in the zero-form sector were presented in Didenko et al. (JHEP 2012:184, 2020) in the Z-dominated form implying their spin-locality by virtue of Z-dominance Lemma of Gelfond and Vasiliev (Phys. Lett. B 786:180, 2018). However the vertex of Didenko et al. (2020) 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 ${{\rm Y}}_{\omega CCC}^{\eta \eta }$ 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.