A Note on a Brill-Noether Locus over a Non-hyperelliptic curve of genus 4

We prove that a certain Brill-Noether locus over a non-hyperelliptic curve $C$ of genus 4, is isomorphic to the $\mathit{\text{Donagi-Izadi}}\mathit{\text{cubic}}\mathit{\text{threefold}}$ in the case when the pencils of the two trigonal line bundles of $C$ coincide.