Computation of the component group of an arbitrary real algebraic group

We compute explicitly the group of connected components $\pi_0G(\mathbb{R})$ of the real Lie group $G(\mathbb{R})$ for an arbitrary (not necessarily linear) connected algebraic group $G$ defined over the field $\mathbb{R}$ of real numbers. In particular, it turns out that $\pi_0G(\mathbb{R})$ is always an elementary Abelian $2$-group. The result looks most transparent in the cases where $G$ is a linear algebraic group or an Abelian variety. The computation is based on structure results on algebraic groups and Galois cohomology methods.