Given an algebraically closed field $\mathbb{F}$ of characteristic $0$ and a closed subgroup $G$ of $GL_n(\mathbb{F})$. I want to find the irreducible component of $1$. Is there a better way then computing the primary decomposition, pick the component, which contains $1$, and compute its radical?

Given an algebraically closed field $\mathbb{F}$ of characteristic $0$ and a closed subgroup $G$ of $GL_n(\mathbb{F})$. Let $\{g_1,\ldots, g_r\}$ be a Gr"obner basis for the correpsonding ideal $\mathcal{I}(G)$. I want to find the irreducible component of $1$ (that means i want to find generators for $\mathcal{I}(G^\circ)$). Is there a better way than computing the primary decomposition, pick the component, which contains $1$, and compute its radical?