Аннотация:For a connected linear algebraic group G defined over the field of real numbers R, we compute the component group of the real Lie group G(R) in terms of a maximal split torus T ⊆ G. In particular, we recover a theorem of Matsumoto (1964) that each connected component of G(R) intersects T(R). We provide explicit elements of T(R) which represent all connected components of G(R). The computation is based on structure results for real loci of algebraicgroups and on methods of Galois cohomology.