= Cubic and quartic Yang-Mills self-interactions
{title2=$\mathcal L_3=gf_{abc}\partial_\mu A_\nu^aA^{b\mu}A^{c\nu},\quad\mathcal L_4=-\frac{g^2}4f_{abc}f_{ade}A_\mu^bA_\nu^cA^{d\mu}A^{e\nu}$}
With $D_\mu=\partial_\mu+igA_\mu$ and $[T^a,T^b]=if_{abc}T^c$, the <Yang-Mills field strength> has the sign $F_{\mu\nu}^a=G_{\mu\nu}^a-gf_{abc}A_\mu^bA_\nu^c$. Expanding $-F^a_{\mu\nu}F^{a\mu\nu}/4$ gives a cross term $(g/2)G^a_{\mu\nu}f_{abc}A^{b\mu}A^{c\nu}$; 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.
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