though $A$ contains $Q$ and $(A,\sigma)$ decomposes into a tensor product of quaternion algebras with involution. This situation does not occur in lower degrees.

Algebras with involution that become split hyperbolic over $F_Q$ are considered in §\[split.sec\]. It is shown in Proposition \[split.cor\] that their anisotropic kernel contains ${(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}$ (if it is not trivial). This applies in particular to algebras of degree $2m$ with $m$ odd, see Corollary \[split2.cor\]. The case of algebras of degree $4$ is completely elucidated in §\[deg4.sec\], using Clifford algebras for orthogonal involutions and a relative cohomological invariant of degree $3$ due to Knus–Lam–Shapiro–Tignol for symplectic involutions. In the symplectic case, we classify the algebras of degree $4$ that become hyperbolic over $F_Q$ but do not contain ${(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}$, see Theorem \[deg4symp.thm\]. We show in §\[min.sec\] that our result is equivalent to the Hoffmann–Lewis–Van Geel classification of $5$-dimensional quadratic forms that become isotropic over $F_Q$ without containing a Pfister neighbour of the norm form of $Q$, see Corollary \[qf.cor\]. Orthogonal involutions on algebras of degree $8$ are considered in §§\[totdec.sec\] and \[deg8.sec\], using triality. In §\[totdec.sec\] we relate tensor products of quaternion algebras to quadratic forms of dimension $8$ with trivial discriminant. The algebras with involution that 