to ${{\left\langle{1,-a,-b}\right\rangle}}$.

The equivalence of (a), (b), (c) is clear if $A$ is split, since then $(A,\sigma)\simeq\operatorname{Ad}_\varphi$ and $\varphi$ is a $3$-fold Pfister form, hence it is isotropic if and only if it is hyperbolic. For the rest of the proof, we may thus assume $A$ is not split. If $(A,\sigma)$ is hyperbolic, then it follows from [@G:clif] that the split component of $(C(A,\sigma),\underline{\sigma})$ is isotropic, hence (b)$\Rightarrow$(c). Conversely, if (c) holds, then [@KMRT (8.5)] shows that $(C_0(\varphi),\tau_0)$ is hyperbolic, hence $(C_+(\varphi),\tau_+)$ also is hyperbolic, proving (c)$\Rightarrow$(b). The equivalence of (a) and (b) readily follows from [@BFPQM Prop. 2.10].

To give an example of a division $F$-algebra of degree $8$ with a totally decomposable orthogonal involution that is hyperbolic over $F_Q$ but does not contain ${(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}$, we use an example of $F_Q$-minimal quadratic form of dimension $7$ due to Hoffmann and Van Geel [@HVG]. For the rest of this section, we fix the following notation: $F_1=F_0(t,u)$ is the function field in two independent indeterminates over an arbitrary field $F_0$ of characteristic different from $2$, and $F=F_1((a))((b))$ is the iterated Laurent series field in two indeterminates $a$, $b$. In accordance with our running notation, $Q$ denotes the quaternion algebra $(a,b)_F$. Let 