case in the propositions above, we have:

\[split.cor\] Let $(A,\sigma)$ be a central simple $F$-algebra with involution. Assume $\sigma$ is anisotropic and $(A,\sigma)_{F_Q}$ is split hyperbolic. Then either

-   $A$ is split, $\sigma$ is orthogonal, and there is a quadratic form $q$ such that $$(A,\sigma)\simeq\operatorname{Ad}_{n_Q\otimes
        q}\simeq{(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}\otimes{(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}\otimes\operatorname{Ad}_q;$$ or

-   $A$ is Brauer-equivalent to $Q$, $\sigma$ is symplectic, and there is a quadratic form $q$ such that $$(A,\sigma)\simeq{(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}\otimes\operatorname{Ad}_q.$$

In both cases, $(A,\sigma)$ contains ${(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}$.

Using the results in this section, we may describe the algebras with involution of degree $2\bmod4$ that become hyperbolic over $F_Q$:

\[split2.cor\] Let $(A,\sigma)$ be a central simple $F$-algebra with involution. If $\deg A\equiv2\bmod4$ and $(A,\sigma)_{F_Q}$ is hyperbolic, then $A_{F_Q}$ is split. If moreover $\sigma$ is anisotropic, then it is symplectic and there is an odd-dimensional quadratic form $q$ over $F$ such that $(A,\sigma)\simeq{(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}\otimes\operatorname{Ad}_q$.

Since $\deg A\equiv2\bmod4$, we have $A\simeq M_r(H)$ for some odd integer $r$ and some quaternion algebra $H$ over $F$. If $H$ is division, $A$ does not admit a hyperbolic involution. Therefore, the hypothesis that $(A,\sigma)_{F_Q}$ is hyperbolic implies that $H_{F_Q}$ is split, hence $(A,\sigma)_{F_Q}$ is split hyperbolic. The last statement then readily follows from Corollary \[split.cor\].

In view of this corollary, we only consider 