Killing-form Yang-Mills Lagrangian (${\mathcal{L}}_{\mathrm{YM}}$)

For a real compact semisimple algebra, $B=-\kappa$ is a positive internal metric. With signature $(+---)$ and connection $D_{\mu }={\partial }_{\mu }+A_{\mu }$, the healthy Lagrangian is $\mathcal{L}=-B(F_{\mu \nu },F^{\mu \nu })/(4g^2)=\kappa (F_{\mu \nu },F^{\mu \nu })/(4g^2)$. Gauge invariance follows from Killing-form invariance and ${\delta }_{X}F=ϵ[X,F]$. The physical Hamiltonian density is $[B(E_i,E_i)+B(B_i,B_i)]/(2g^2)$ after imposing the Gauss law constraint in gauge theory and treating the spatial boundary flux appropriately. An Abelian factor needs a separate positive invariant metric because its Killing form vanishes.