= Solution
<Gauge boson self-interactions> follow from noncommuting generators, rather than a separate assumption that the force carriers should attract one another. With $D_\mu=\partial_\mu+igA_\mu^aT^a$ and $[T^a,T^b]=if^{abc}T^c$, the <Yang-Mills field strength> is
$$
F_{\mu\nu}^a=\partial_\mu A_\nu^a-\partial_\nu A_\mu^a-gf^{abc}A_\mu^bA_\nu^c.
$$
The last term is required so that $F_{\mu\nu}$ transforms covariantly under the <gauge group>. Let $G_{\mu\nu}^a=\partial_\mu A_\nu^a-\partial_\nu A_\mu^a$. Expanding $\mathcal L=-F_{\mu\nu}^aF^{a\mu\nu}/4$ gives a quadratic kinetic term and the <cubic and quartic Yang-Mills self-interactions>
$$
\mathcal L_3=gf^{abc}(\partial_\mu A_\nu^a)A^{b\mu}A^{c\nu},\qquad
\mathcal L_4=-\frac{g^2}{4}f^{abc}f^{ade}A_\mu^bA_\nu^cA^{d\mu}A^{e\nu}.
$$
The <structure constants> and one <gauge coupling> fix both types of vertex. For an Abelian group the structure constants vanish: the renormalizable pure <electromagnetic field> action has no elementary three- or four-<photon> vertex. Photon scattering through charged-particle loops is a different statement, not a contradiction.
In <Quantum chromodynamics>, the non-Abelian $SU(3)_c$ structure produces three- and four-<gluon> vertices; the gluons themselves carry <color charge>. These vertices influence multi-jet production and the <renormalization group> running of the strong coupling. In the electroweak sector the $SU(2)_L$ fields mix with the Abelian field after <electroweak symmetry breaking>. The resulting charged-vector vertices include $W^+W^-\gamma$ and $W^+W^-Z$, with couplings $e$ and $g\cos\theta_W$, together with the related four-boson vertices. Neutral-only triple vertices such as $ZZ\gamma$ are absent at tree level in the <Standard Model>.
The <electroweak triple gauge coupling> can be measured using $e^+e^-\to W^+W^-$. The neutrino exchange and neutral-vector exchange diagrams interfere: the gauge-related coefficients cancel the most rapidly growing longitudinal-polarization terms. Their relative strengths and angular dependence therefore test the theory more sharply than merely observing a pair of massive vector bosons. The electron-mass-suppressed Higgs contribution completes the corresponding mass-dependent cancellation. Adding arbitrary independent three- and four-vector couplings would generally destroy these relations.
At the <Large Electron–Positron Collider>, the <L3 experiment> fitted charged triple-gauge parameters using hadronic and semileptonic W-pair events, complemented by single-W and single-photon channels. These measurements agreed with the predicted self-couplings. Their interpretation uses reconstructed decay products and differential distributions, rather than direct pictures of a three-boson vertex.<Measurement of Triple-Gauge-Boson Couplings of the W Boson at LEP by L3 Collaboration>[2004, abstract: W-pair channels at 189–209 GeV, complementary channels and agreement with Standard Model self-couplings]{ref}
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