is similar to a Pfister neighbour of ${{\langle\!\langle{a',b',\lambda}\rangle\!\rangle}}$, by [@KMRT p. 271]. Thus, (a) and (b) hold if $\varphi$ is $F_Q$-minimal. Conversely, if $C_0(\varphi)\simeq M_2(Q')$ for some quaternion $F$-algebra $Q'=(a',b')_F$, then as observed in the proof of Proposition \[split.prop\](2) we have $$(C_0(\varphi),\tau_0)\simeq\operatorname{Ad}_{{{\langle\!\langle{\mu}\rangle\!\rangle}}}\otimes(Q',{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})
  \qquad\text{for some $\mu\in F^\times$}.$$ It then follows from [@KMRT p. 271] that $\varphi$ is similar to a Pfister neighbour of ${{\langle\!\langle{a',b',\mu}\rangle\!\rangle}}$. If (a) holds, then ${{\langle\!\langle{a,b,\lambda}\rangle\!\rangle}}\simeq{{\langle\!\langle{a',b',\mu}\rangle\!\rangle}}$ and by the common slot lemma we may assume $\lambda=\mu$. Thus, $\varphi$ is $F_Q$-minimal if (a) and (b) hold.

Totally decomposable orthogonal involutions of degree 8 {#totdec.sec}
=======================================================

In this section, $A$ denotes a central simple $F$-algebra of degree $8$ and $\sigma$ is an orthogonal involution on $A$. The algebra with involution $(A,\sigma)$ is called *totally decomposable* if there are $\sigma$-stable quaternion subalgebras $Q_1$, $Q_2$, $Q_3$ in $A$ such that $A=Q_1\otimes Q_2\otimes
Q_3$. Denoting by $\sigma_i$ the restriction of $\sigma$ to $Q_i$ for $i=1$, $2$, $3$, we then have $$(A,\sigma)=(Q_1,\sigma_1)\otimes(Q_2,\sigma_2)\otimes(Q_3,\sigma_3).$$ The totally decomposable algebras of degree $8$ are characterized by the property that $\operatorname{disc}\sigma=1$ and one of the components of the Clifford algebra $C(A,\sigma)$ is split, see [@KMRT (42.11)]. Proposition \[totdec.prop\] below shows how to use this criterion to relate totally decomposable 