Isometries of Mukai lattices

A Mukai lattice is a free finitely generated $\mathbb{Z}$-module equipped with a unimodular not necessarily symmetric bilinear form. A standard example of a Mukai lattice is the Grothendieck group of an algebraic variety $X$ equipped with the so-called Euler form. We will talk about the group of isometries of this lattice in particular case when $X$ is a complex projective space. It turns out that this group has a nice structure; for instance, we will see that it is essentially isomorphic to the free abelian group of rank$ [\frac{n+1}{2}].$ We will also compute explicitly its generators for all $n$ not exceeding 6.