  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 $T_{\text{s}}\subseteq G$. In
particular, we recover a theorem of Matsumoto (1964) that each connected
component of $G(\mathbb{R})$ intersects $T_{\text{s}}(\mathbb{R})$. We provide
explicit elements of $T_{\text{s}}(\mathbb{R})$ 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.
