Let Ea(u,v) be the extremal algebra determined by two hermitians u and v with u^2=v^2=1. We show that: Ea(u,v)={f+gu:f,g\in C(𝕋)} , where [ ] is the unit circle; Ea(u,v) is C^*-equivalent to C^*({\cal G}), where {\cal G} is the infinite dihedral group; most of the hermitian elements k of Ea(u,v) have the property that k^n is hermitian for all odd n but for no even n; any two hermitian words in {\cal G} generate an isometric copy of Ea(u,v) in Ea(u,v).