The connected components of a finite-type algebraic variety are finitely many, closed, and coincide with irreducible components after taking the component containing a smooth point such as \(e\). Multiplication sends the product of two connected subsets containing \(e\) into a connected subset containing \(e\), hence \(G^\circ G^\circ \subseteq G^\circ\); inversion similarly preserves \(G^\circ\). Thus \(G^\circ\) is a closed subgroup.

For any \(g \in G\), conjugation by \(g\) is an algebraic automorphism of \(G\) and sends \(e\) to \(e\), so it preserves the identity component. Therefore \(G^\circ\) is normal. Since \(G\) has only finitely many connected components, \(G/G^\circ\) is finite. Finally, if \(H \subseteq G\) is closed of finite index, then \(G/H\) is finite and the image of the connected set \(G^\circ\) in it is a point; since \(e \in H\), this gives \(G^\circ \subseteq H\).