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On the Continuous Cohomology of a semi-direct product Lie group

Let $G$ be a Lie group and $H$ be a subgroup of it. We can construct a bisimplicial manifold $NG(*) \rtimes NH(*)$ and the de Rham complex $Ω^*(NG(*) \rtimes NH(*))$ on it. This complex is a triple complex and the cohomology of its total complex is isomorphic to $H^*(B(G \rtimes H))$. In this paper, we show that the total complex of the double complex $Ω^q(NG(*) \rtimes NH(*))$ is isomorphic to the continuous cohomology $H_c^*(G \rtimes H;S^q{\mathcal G}^* \otimes S^q{\mathcal H}^*)$ for any fixed $q$.