= Killing-form Lagrangian for an adjoint scalar
{c}
An adjoint scalar has <adjoint covariant derivative> $D_\mu\phi=\partial_\mu\phi+[A_\mu,\phi]$. A symmetric <invariant bilinear form on a Lie algebra> contracts the <gauge field strength> and scalar kinetic terms to give <gauge invariance>. On a <compact real form> the positive internal metric is $-\kappa$, because the <Killing form> itself is negative definite; complex scalar components require the associated Hermitian pairing. The Killing form is degenerate on Abelian factors, where another invariant metric is needed for a nondegenerate kinetic term.
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