This question is related to requirement that the gauge group of a gauge theory be a direct product of compact simple groups and $U(1)$ factors but is not the same as, for example, this question (though related as described below).
When looking to build the kinetic term of a gauge theory, the demand that the Lagrangian be real and Lorentz invariant implies the lowest order term we can write down using only the field strength $F^{\alpha}_{\mu\nu}$ (using $\alpha,\beta,\ldots$ for gauge group indices) is of the form $$ g_{\alpha\beta}F^{\alpha}_{\mu\nu}F^{\beta\mu\nu} $$ where $g_{\alpha\beta}$ is a real matrix which we may take to be symmetric.
As described in this answer and Weinberg Vol2, in order to conclude that the gauge group must be a product of compact simple and $U(1)$ factors we must ague both that $g_{\alpha\beta}$ satisfy the invariance condition $g_{\alpha\beta}C^\beta_{\gamma\delta}=-g_{\gamma\beta}C^\beta_{\alpha\delta}$ (so $g_{\alpha\beta}$ is proportional to the Killing form of the gauge group) and also that $g_{\alpha\beta}$ must be positive-definite.
The former of these follows from gauge invariance and does not concern me here. My question: The claim is that the positive definiteness of $g_{\alpha\beta}$ follows from unitarity and the canonical quantization procedure. Can anyone make explicit how this follows?