Let $G$ be an algebraic group over $\mathbb{C}$. Suppose $G$ is given an $\mathbb{R}$-structure, i.e. an $\mathbb{R}$-algebra $A_0$ such that $A = \mathbb{C} \otimes_{\mathbb{R}} A_0$, where $A$ is the coordinate ring of $G$. Then the Galois group $\Gamma = \textrm{Gal}(\mathbb{C}/\mathbb{R})$ acts as a group of ring automorphisms $A$ as $\sigma.(\lambda \otimes a) = \sigma(\lambda) \otimes a$, which induces an action of $\Gamma$ on $\textrm{Spm } A = G$.

Then $\Gamma$ further acts on $\textrm{Aut}(G)$, the group of algebraic group automorphisms of $G$, as $\gamma.\phi(x) = \gamma^{-1}.\phi(\gamma.x)$

The isomorphism classes of $\mathbb{R}$-structures on $G$ form a pointed set, the distinguished element corresponding to class of $A_0$. This pointed set is isomorphic to $H^1(\Gamma, \textrm{Aut}(G))$, the set of equivalence classes of one-cocycles, i.e. functions $c: \Gamma \rightarrow \textrm{Aut}(G)$ for which $c(\gamma_1\gamma_2) = c(\gamma_1) \circ \gamma_1.(c(\gamma_2))$ for all $\gamma_i \in \Gamma$.

If $M$ is a symmetric invertible real matrix, one can define $\textrm{O}_n(\mathbb{C}) = \{ x \in \textrm{GL}_n(\mathbb{C}) : x^t Mx = M \}$. As an algebraic group over $\mathbb{C}$, $O_n(\mathbb{C})$ does not depend on the choice of $M$ up to isomorphism. But over $\mathbb{R}$, different choice of $M$ may yield different groups.

Is there is a connection between the set $H^1(\Gamma, \textrm{Aut}(G))$ and classes of invertible symmetric matrices $M$? (two matrices $M_1, M_2$ being equivalent if there exists an $x \in \textrm{GL}_n(\mathbb{R})$ such that $x^tM_1x = M_2$)