Semiorthogonal decompositions and components of derived categories of orthogonal Grassmannian fibrations

Kuznetsov showed that for a flat quadric fibration $\mathcal{Q}$ over a smooth base $S$, ${\mathrm{D}}^b(\mathcal{Q})$ admits a semiorthogonal decomposition where one of the components is the derived category of the sheaf of even parts of a Clifford algebra ${\mathrm{D}}^b(S,\mathcal{C}{l}_0)$. \par As progress towards a generalization, we show that for a quadric fibration with a fairly minor condition on the rank of the quadric fibers, the category ${\mathrm{D}}^b(S,\mathcal{C}{l}_0)$ embeds fully faithfully into the derived category of the relative orthogonal Grassmannian ${\mathrm{D}}^b(\mathrm{O}\mathrm{G}\mathrm{r}(k,\mathcal{Q}))$. When $k=2$, we use this to produce a semiorthogonal decomposition of ${\mathrm{D}}^b(\mathrm{O}\mathrm{G}\mathrm{r}(2,\mathcal{Q}))$ up to a residual category; we compute this residual category in the smooth case and produce a conjecture for in the case of a pencil of quadrics with smooth base locus.