Geodesic
Computation of the component group of an arbitrary real algebraic group
Fundamentalʹnaâ i prikladnaâ matematika, Tome 24 (2023) no. 4, pp. 199-211.

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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. 
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D. A. Timashev. Computation of the component group of an arbitrary real algebraic group. Fundamentalʹnaâ i prikladnaâ matematika, Tome 24 (2023) no. 4, pp. 199-211. http://geodesic.mathdoc.fr/item/FPM_2023_24_4_a10/

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