On the Component Group of a Real Algebraic Group

For a connected linear algebraic group $G$ defined over $\mathbb R$, we compute the component group $\pi _0G(\mathbb R)$ of the real Lie group $G(\mathbb R)$ in terms of a maximal split torus $T_{\mathrm{s}}\subseteq G$. In particular, we recover a theorem of Matsumoto (1964) that each connected component of $G(\mathbb R)$ intersects $T_{\mathrm{s}}(\mathbb R)$. We provide explicit elements of $T_{\mathrm{s}}(\mathbb R)$ which represent all connected components of $G(\mathbb R)$. The computation is based on structure results for real loci of algebraic groups and on methods of Galois cohomology.