  According to Mukai, any prime Fano threefold X of genus 7 is a linear section
of the spinor tenfold in the projectivized half-spinor space of Spin(10). It is
proven that the moduli space of stable rank-2 vector bundles with Chern classes
c_1=1,c_2=5 on a generic X is isomorphic to the curve of genus 7 obtained by
taking an orthogonal linear section of the spinor tenfold. This is an inverse
of Mukai's result on the isomorphism of a non-abelian Brill--Noether locus on a
curve of genus 7 to a Fano threefold of genus 7. An explicit geometric
construction of both isomorphisms and a similar result for K3 surfaces of genus
7 are given.
