${(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}$, see §\[deg4orth.sec\]. For algebras of degree $8$, case 1 is when $(A,\sigma)$ is totally decomposable, see §\[totdec.sec\]; case 2 is considered in §\[deg8.sec\].

Algebras with involution that become split hyperbolic over $F_Q$ {#split.sec}
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Since $F_Q$ is the function field of the Severi–Brauer variety of $Q$, the Brauer group kernel of the scalar extension from $F$ to $F_Q$ is $\{0,[Q]\}$. Therefore, if $(A,\sigma)$ is a central simple $F$-algebra with involution that becomes split hyperbolic over $F_Q$, then either $A$ is split or $A$ is Brauer-equivalent to $Q$. We consider each case separately.

\[split.prop\] Let $(A,\sigma)$ be a split central simple $F$-algebra with involution.

1.  If $\sigma$ is symplectic, then it is hyperbolic. In this case, $(A,\sigma)$ contains ${(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}$ if and only if $\deg A\equiv0\bmod4$.

2.  If $\sigma$ is orthogonal, then $(A,\sigma)\simeq\operatorname{Ad}_q$ for some quadratic form $q$. We have $(\operatorname{Ad}_q)_{F_Q}$ hyperbolic if and only if the anisotropic kernel of $q$ is a multiple of the norm form $n_Q$. Assuming this condition holds, $\operatorname{Ad}_q$ contains ${(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}$ if and only if the Witt index of $q$ is a multiple of $4$.

\(1) Since every alternating form on an $F$-vector space is hyperbolic, it follows that every split algebra with hyperbolic involution is 