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For any connected linear algebraic group $G$ over your field $K=\mathbf{Q}(\sqrt{-3})$, the group of $K$-points $G(K)$ is dense in $G(\mathbf{C})$ for the complex topology. This follows from the real approximation theorem for connected linear algebraic groups, see a reference to Sansuc's paper in my comments to A question on algebraic torus.