Algebraically primitive Teichmüller curves in Ω\mathcal M_g(2g-2)^{$\mathrm{h}\mathrm{y}\mathrm{p}$}

We give a complete classification of algebraically primitive Teichmüller curves in $\mathrm{\Omega }{\mathcal{M}}_g(2g-2{)}^{\mathrm{h}\mathrm{y}\mathrm{p}}$ for $g\ge 5$. They are precisely the curves generated by the double regular $(2g+1)$-gon when $2g+1$ is prime, and by the regular $4g$-gon when $g$ is a power of two.