Remark

Consider the subcomplexes $Z(\Omega (G))$ and $B(\Omega (G))$ of $\Omega (G) ($Algèbre, Chap. X, p. 25). It follows from the formula giving the differential of the product of two forms (Differentiable and Analytic Manifolds, Results, 8.3.5) that $Z(\Omega (G))$ is a subalgebra of $\Omega (G)$ and that $B(\Omega (G))$ is an ideal of $Z(\Omega (G))$; consequently, the exterior product induces a graded algebra structure on $H(\Omega (G))$. The preceding now gives an isomorphism of graded algebras $H(\Omega (G))\rightarrow$ Alt($\mathfrak{g}$)$^G$.

Let H be a closed subgroup of G; we apply Th. 2 to $X = G/H$. By Chap. III, §1, no. 8, Cor. 1 of Prop. 17, the G-invariant differential forms on $G/H$ can be identified with the H-invariant elements of Alt(T$_e(G/H)$), that is with the elements of Alt($\mathfrak{g}$) that are H-invariant and annihilated by the operators $i(\xi )$ for all $\xi \in L(H)$. Consequently: