Computing Galois cohomology of a real linear algebraic group

Abstract

Let $G$ be a linear algebraic group, not necessarily connected or reductive, over the field of real numbers $R$. We describe a method, implemented on computer, to find the first Galois cohomology set $H^1(R,G)$. The output is a list of 1-cocycles in $G$. Moreover, we describe an implemented algorithm that, given a 1-cocycle $z\in Z^1(R,G)$, finds the cocycle in the computed list to which $z$ is equivalent, together with an element of $G\left(C\right)$ realizing the equivalence.