of vector bundles on $X$.

Recall here the definition of a full strong exceptional sequence. An object $G_i$ of the bounded derived category $D^b(X)$ of coherent sheaves on $X$ is said to be exceptional if $\operatorname{RHom}(G_i, G_i)\cong K$ and a sequence $G_0,\dots,G_m$ of exceptional objects is said to be exceptional if $\operatorname{RHom}(G_i, G_j)=0$ for all $0\leqq j<i\leqq m$. An exceptional sequence $G_0,\dots,G_m$ is said to be strong if $\operatorname{Ext}^k(G_i, G_j)=0$ for all $k>0$ and $0\leqq i<j\leqq m$. Finally a strong exceptional sequence $G_0,\dots,G_m$ is said to be full if $D^b(X)$ is the smallest triangulated full subcategory containing $G_0,\dots,G_m$ and closed under isomorphism.

If $X$ is an $n$-dimensional projective space $\mathbb{P}^n$, then $(\mathcal{O},\mathcal{O}(1),\dots,\mathcal{O}(n))$ is a strong exceptional sequence of line bundles, and it is full by Beilinson’s theorem [@MR0509388 Theorem]. If $X$ is an odd-dimensional smooth hyperquadric $\mathbb{Q}^n$ and the characteristic $\operatorname{char}K$ of the base field $K$ is zero, then it follows from Bott’s vanishing theorem that $(\mathcal{O},\mathcal{S},\mathcal{O}(1),\dots,\mathcal{O}(n-1))$ is a strong exceptional sequence of vector bundles, where $\mathcal{S}$ is the spinor bundle, and it is full by Kapranov’s theorem [@MR0939472 Theorem 4.10]. If $X$ is an even-dimensional smooth hyperquadric $\mathbb{Q}^n$ and $\operatorname{char}K=0$, then Bott’s vanishing theorem shows that $(\mathcal{O},\mathcal{S}^+,\mathcal{S}^-,\mathcal{O}(1),\dots,\mathcal{O}(n-1))$ is a 