An LLL algorithm with symmetries

We give a generalisation of the Lenstra-Lenstra-Lov\'asz (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 $G = \mathrm{Sp}_{2g}$, $\mathrm{SO}_{2g}$, and for the exceptional group $G = G_2$.