do not decompose into tensor products of quaternion algebras with involution and become hyperbolic over $F_Q$ are determined in §\[deg8.sec\], see Theorem \[deg8.thm\]. They are isotropic, and their anisotropic kernel contains ${(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}$. Finally, we use Laurent power series in §\[large.sec\] to construct algebras with involution of large degree that do not contain ${(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}$.

Notations and preliminary observations
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We work over a base field $F$ of characteristic different from $2$, and only consider algebras with involution of the first kind. We refer the reader to [@KMRT] and to [@Lam] for background information on central simple algebras with involution and on quadratic forms. However, we depart from the notation in [@Lam] by using ${{\langle\!\langle{a_1,\dots, a_n}\rangle\!\rangle}}$ to denote the $n$-fold Pfister form $\otimes_{i=1}^n{{\left\langle{1,-a_i}\right\rangle}}$.

If $h\colon V\times V\to D$ is a regular hermitian or skew-hermitian form on a finite-dimensional vector space $V$ over a division algebra $D$, we denote by $\operatorname{ad}_h$ the involution on $\operatorname{End}_D(V)$ that is adjoint to $h$. In the particular case where $D$ is split and $h$ is the polar form of a quadratic form $q$, we also denote $\operatorname{ad}_q$ for $\operatorname{ad}_h$. Following Becher [@Becher], for any $n$-dimensional quadratic form $q$ over $F$, we denote by $\operatorname{Ad}_q$ the split orthogonal 