Suppose that $G$ is a complex, reductive algebraic group. A real form of $G$ is an antiholomorphic involutive automorphism $\sigma$, so $G\left(\mathbb{R}\right)=G(\mathbb{C}{)}^{\sigma }$ is a real Lie group. Write $H^1(\sigma ,G)$ for the Galois cohomology (pointed) set $H^1(Gal(\mathbb{C}/\mathbb{R}),G)$. A Cartan involution for $\sigma$ is an involutive holomorphic automorphism $\theta$ of $G$, commuting with $\sigma$, so that $\theta \sigma$ is a compact real form of $G$. Let $H^1(\theta ,G)$ be the set $H^1({\mathbb{Z}}_2,G)$, where the action of the nontrivial element of ${\mathbb{Z}}_2$ is by $\theta$. By analogy with the Galois group, we refer to $H^1(\theta ,G)$ as the Cartan cohomology of $G$ with respect to $\theta$. Cartan’s classification of real forms of a connected group, in terms of their maximal compact subgroups, amounts to an isomorphism $H^1(\sigma ,G_{\mathrm{ad}})\simeq H^1(\theta ,G_{\mathrm{ad}})$, where $G_{\mathrm{ad}}$ is the adjoint group. Our main result is a generalization of this: there is a canonical isomorphism $H^1(\sigma ,G)\simeq H^1(\theta ,G)$.

We apply this result to give simple proofs of some well-known structural results: the Kostant–Sekiguchi correspondence of nilpotent orbits; Matsuki duality of orbits on the flag variety; conjugacy classes of Cartan subgroups; and structure of the Weyl group. We also use it to compute $H^1(\sigma ,G)$ for all simple, simply connected groups and to give a cohomological interpretation of strong real forms. For the applications it is important that we do not assume that $G$ is connected.