A criterion for quadraticity of a representation of the fundamental group of an algebraic variety

Let $\mathrm{\Gamma }$ be a finitely presented group and $G$ a linear algebraic group over $\mathbb{R}$. A representation $\rho :\mathrm{\Gamma }\to G(\mathbb{R})$ can be seen as an $\mathbb{R}$-point of the representation variety $\mathfrak{R}(\mathrm{\Gamma },G)$. It is known from the work of Goldman and Millson that if $\mathrm{\Gamma }$ is the fundamental group of a compact K{"a}hler manifold and $\rho$ has image contained in a compact subgroup then $\rho$ is analytically defined by homogeneous quadratic equations in $\mathfrak{R}(\mathrm{\Gamma },G)$. When $X$ is a smooth complex algebraic variety, we study a certain criterion under which this same conclusion holds.