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We describe a general (primitively) polarized K3 surface $(S,h)$ with$(h^2)=24$ as a complete intersection variety with respect to vector bundles onthe $6$-dimensional moduli space $\mathcal{N}^-$ of the stable vector bundlesof rank two with fixed odd determinant on a curve $C$ of genus $3$. If thecurve $C$ is hyperelliptic, then $\mathcal{N}^-$ is a subvariety of the$12$-dimensional Grassmann variety $\operatorname{Gr}(\mathbf{C}^8,2)$ definedby a pencil of quadric forms. In this case, our description implies that ageneral $(S,h)$ is the intersection of two (7-dimensional) contact homogeneousvarieties of $\operatorname{Spin}(7)$ in the Grassmann variety$\operatorname{Gr}(\mathbf{C}^8,2)$.

Polarized K3 surfaces of genus thirteen and curves of genus three