A Lemma on Leech-like Lattices

A Leech pair is defined as a pair $(G,S)$, where $S$ is a positive definite even lattice without roots, equipped with a faithful action of a finite group $G$, such that the invariant sublattice of $S$ under the action of $G$ is trivial, and the induced action of $G$ on the discriminant group of $S$ is also trivial. This structure appears naturally when investigating hyperk\"ahler manifolds and the symplectic automorphisms acting on them. An important lemma due to Gaberdiel--Hohenegger--Volpato asserts that a Leech pair $(G,S)$ admits a primitive embedding into the Leech lattice if $rank(S)+\ell(A_S)\le 24$. However, the original proof is incomplete, as demonstrated by a counterexample provided by Marquand and Muller. They also presented a computer-assisted proof of the lemma for cases where $rank(S) \le 21$. In this paper, we modify the original approach to provide a complete and conceptual proof of the lemma.