Computation of the Component Group of an Arbitrary Real Algebraic Group

We compute explicitly the group of connected components π₀G( \({\mathbb{R}}\) $\mathbb{R}$ ) of the real Lie group G( \({\mathbb{R}}\) $\mathbb{R}$ ) for an arbitrary (not necessarily linear) connected algebraic group G defined over the field \({\mathbb{R}}\) $\mathbb{R}$ of real numbers. In particular, it turns out that π₀G( \({\mathbb{R}}\) $\mathbb{R}$ ) 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.