Algebraic Families of Groups and Commuting Involutions

Let $G$ be a complex affine algebraic group, and let ${\sigma }_1$ and ${\sigma }_2$ be commuting anti-holomorphic involutions of $G$. We construct an algebraic family of algebraic groups over the complex projective line and a real structure on the family that interpolates between the real forms $G^{{\sigma }_1}$ and $G^{{\sigma }_2}$.