= Fayet–Iliopoulos terms require an Abelian gauge factor
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{title2=$\xi_a f^{a}{}_{bc}=0$}
A constant linear auxiliary-field term $\xi_aD^a$ is <gauge-invariant> only when $\xi$ is an invariant vector in the dual adjoint representation. Equivalently, it annihilates all Lie brackets. A simple non-Abelian <Lie algebra> equals its commutator algebra and has no such nonzero vector. Abelian factors do admit the term. Identifying a Fayet–Iliopoulos parameter in a simple non-Abelian theory therefore requires setting it to zero, rather than treating it as an arbitrary coupling.
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