An LLL algorithm with symmetries

We give a generalisation of the Lenstra-Lenstra-Lovász (LLL) lattice-reduction algorithm that is valid for an arbitrary (split, semisimple) reductive group $G$. This can be regarded as `lattice reduction with symmetries’. We make this algorithm explicit for the classical groups $G={\mathrm{S}\mathrm{p}}_{2g}$, ${\mathrm{S}\mathrm{O}}_{2g}$, and for the exceptional group $G=G_2$.