Let $(A,\sigma)$ be a central simple $F$-algebra with orthogonal involution and $\deg A\equiv0\bmod4$. If $(A,\sigma)_{F_Q}$ is hyperbolic, then $\operatorname{disc}\sigma=1$ and at least one of $(C_+(A,\sigma),\sigma_+)$, $(C_-(A,\sigma),\sigma_-)$ is split and isotropic.

Since $\sigma_{F_Q}$ is hyperbolic, $\operatorname{disc}\sigma$ is a square in $F_Q$, hence also in $F$ since $F$ is quadratically closed in $F_Q$. By [@KMRT (8.31)], the Clifford algebra of any hyperbolic involution has a split component; therefore $C_+(A,\sigma)_{F_Q}$ or $C_-(A,\sigma)$ is split. The corresponding involution $\sigma_\pm$ is isotropic over $F_Q$ by the main theorem in [@G:clif].

Since the Brauer group kernel of the scalar extension map from $F$ to $F_Q$ is $\{0,[Q]\}$, the description of algebras $(A,\sigma)$ with $\sigma$ orthogonal and $\deg A\equiv0\bmod4$ such that $(A,\sigma)_{F_Q}$ is hyperbolic falls into two cases:

Case 1

:   $\bigl\{[C_+(A,\sigma)],[C_-(A,\sigma)]\bigr\}=
    \bigl\{0,[A]\bigr\}$;

Case 2

:   $\bigl\{[C_+(A,\sigma)],[C_-(A,\sigma)]\bigr\}=
    \bigl\{[Q],[A\otimes_FQ]\bigr\}$.

Following Garibaldi’s definition in [@G:16], case 1 is when $(A,\sigma)\in I^3$. Note that these two cases are not exclusive: they intersect if and only if $A$ is Brauer-equivalent to $Q$; then $(A,\sigma)$ becomes split hyperbolic over $F_Q$. This case is easily dealt with in §\[split.sec\]. If $\deg A=4$, the first case arises if and only if $(A,\sigma)$ is hyperbolic, and the second case if and only if $(A,\sigma)$ contains 