Proposition 6

Every compact semi-simple real Lie algebra of rank 1 is isomorphic to $\mathfrak{s}\mathfrak{u}(2,\mathbf{C})$ and to $\mathfrak{o}(3,\mathbf{R})$. Every connected semi-simple compact Lie group of rank 1 is isomorphic to $\mathbf{S}\mathbf{U}(2,\mathbf{C})$ if it is simply-connected, and to $\mathbf{S}\mathbf{O}(3,\mathbf{R})$ if not.

The first assertion follows from the Cor. of Prop. 4 and Prop. 5. Since $\mathbf{S}\mathbf{U}(2,\mathbf{C})$ is homeomorphic to $\mathbf{S}_3($General Topology, Chap. VIII, §1, no. 4, Remark 4), hence is simply-connected (General Topology, Chap. XI, in preparation), every simply-connected compact semi-simple Lie group of rank 1 is isomorphic to $\mathbf{S}\mathbf{U}(2,\mathbf{C})$; every connected compact semi-simple Lie group of rank 1 that is not simply-connected is isomorphic to a quotient of $\mathbf{S}\mathbf{U}(2,\mathbf{C})$ by a subgroup of Z that does not reduce to the identity element, hence to $\mathbf{S}\mathbf{O}(3,\mathbf{R})$.