  We construct a new solution to the tetrahedron equation by further pursuing
the quantum cluster algebra approach in our previous works. The key ingredients
include a symmetric butterfly quiver attached to the wiring diagrams for the
longest element of type $A$ Weyl groups and the implementation of quantum
$Y$-variables through the $q$-Weyl algebra. The solution consists of four
products of quantum dilogarithms, incorporating a total of seven parameters. By
exploring both the coordinate and momentum representations, along with their
modular double counterparts, our solution encompasses various known
three-dimensional (3D) $R$-matrices. These include those obtained by
Kapranov-Voevodsky ('94) utilizing the quantized coordinate ring,
Bazhanov-Mangazeev-Sergeev ('10) from a quantum geometry perspective,
Kuniba-Matsuike-Yoneyama ('23) linked with the quantized six-vertex model, and
Inoue-Kuniba-Terashima ('23) associated with the Fock-Goncharov quiver. The 3D
$R$-matrix presented in this paper offers a unified perspective on these
existing solutions, coalescing them within the framework of quantum cluster
algebra.
