Remark 3

Every point of the group $S_2$ has a neighbourhood homeomorphic to $\mathbf{R}^3$ (Chapter VI, § 2, no. 4, Proposition 5) but $S_4$ is not locally isomorphic to $\mathbf{R}^3$; for if it were it would be abelian, since it is connected (Chapter VII, § 2, no. 2, Theorem 1), and this is not the case, since $i$ and $j$ belong to $S_3$ and $ij \neq ji$ (cf. Chapter V, § 3).