Cubic and quartic Yang-Mills self-interactions (${\mathcal{L}}_3=g{f}_{abc}{\partial }_{\mu }{A}_{\nu }^a{A}^{b\mu }{A}^{c\nu },{\mathcal{L}}_4=-\frac{g^2}{4}{f}_{abc}{f}_{ade}{A}_{\mu }^b{A}_{\nu }^c{A}^{d\mu }{A}^{e\nu }$)

With $D_{\mu }={\partial }_{\mu }+ig{A}_{\mu }$ and $[T^a,T^b]=i{f}_{abc}{T}^c$, the Yang-Mills field strength has the sign $F_{\mu \nu }^a=G_{\mu \nu }^a-g{f}_{abc}{A}_{\mu }^b{A}_{\nu }^c$. Expanding $-F_{\mu \nu }^a{F}^{a\mu \nu }/4$ gives a cross term $(g/2)G_{\mu \nu }^a{f}_{abc}{A}^{b\mu }{A}^{c\nu }$; antisymmetry makes its two derivative terms equal, proving the cubic expression. Squaring the nonlinear term gives the quartic expression. Both vertices arise from the same gauge coupling, rather than independent free parameters.