  We prove that a general cubic in the Hassett divisor $\mathcal{C}_{14}$ of
special cubic fourfolds of discriminant $14$ contains a non-minimal K3 surface
of degree $10$ containing two skew $(-1)$-lines and contained in a smooth
quadric hypersurface $Q^4\subseteq \mathbb{P}^5$, but not contained in any
other (possibly of lower rank) quadric hypersurface.
