Classification of Symplectic Birational Involutions of Manifolds of $OG10$ type

We give a complete classification of symplectic birational involutions of manifolds of $OG10$ type. We approach this classification with three techniques – via involutions of the Leech lattice, via involutions of cubic fourfolds and finally lattice enumeration via a modified Kneser’s neighbour algorithm. The classification consists of three involutions with an explicit geometric realisation via cubic fourfolds, and three exceptional involutions which cannot be obtained by any known construction.