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