dimension $8$ such that $$(C_+(A,\sigma),\sigma_+)\simeq\operatorname{Ad}_\varphi.$$ By triality (see [@KMRT (42.3)]) we have $$(C_0(\varphi),\tau_0)\simeq(A,\sigma)\times(A,\sigma).$$ Therefore, $\operatorname{disc}\varphi=1$ and $(A,\sigma)\simeq
  (C_+(\varphi),\tau_+)$. Conversely, triality also shows that if $(A,\sigma)\simeq(C_+(\varphi),\tau_+)$, then the canonical involution $\underline{\sigma}$ on $C(A,\sigma)$ satisfies $$(C(A,\sigma),\underline{\sigma})\simeq\operatorname{Ad}_\varphi\times(A,\sigma),$$ hence the form $\varphi$ is uniquely determined up to similarity.

The Clifford algebra of $\varphi$ splits if and only if $\varphi$ is a multiple of a $3$-fold Pfister form, by [@Scha Ch. 2, Th. 14.4] and [@Lam X(5.6)]. When that condition holds we have $(A,\sigma)\simeq\operatorname{Ad}_\varphi$ by [@KMRT (35.1)].

If $\varphi$ contains a multiple of ${{\left\langle{1,-a,-b}\right\rangle}}$, then the same argument as in the proof of Proposition \[qf.prop\](2) shows that $(C_0(\varphi),\tau_0)$ contains ${(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}$. Projecting on each component, it follows that $(C_+(\varphi),\tau_+)$ contains ${(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}$. Conversely, if $(A,\sigma)$ contains ${(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}$, then we have $$\label{eq:2}
    (A,\sigma)={(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}\otimes(A_1,\sigma_1)$$ for some central simple algebra with symplectic involution $(A_1,\sigma_1)$ of degree $4$. By [@KMRT (15.19)] there is a $5$-dimensional quadratic form $\psi$ such that $\operatorname{disc}\psi=1$ and $(A_1,\sigma_1)\simeq(C_0(\psi),\tau_0)$. Letting $\tau'_0$ be the canonical involution on $C_0({{\left\langle{ab,-a,-b}\right\rangle}})$, we may rewrite as $$\label{eq:1}
    (A,\sigma)\simeq(C_0({{\left\langle{ab,-a,-b}\right\rangle}}),\tau'_0)\otimes
    (C_0(\psi),\tau_0).$$ In view of the canonical embedding $$C_0({{\left\langle{ab,-a,-b}\right\rangle}})\otimes_FC_0(\psi)\hookrightarrow
  C_0({{\left\langle{ab,-a,-b}\right\rangle}}\perp\psi),$$ which is compatible with the canonical involutions, yields $$(A,\sigma)\simeq(C_+({{\left\langle{ab,-a,-b}\right\rangle}}\perp\psi),\tau_+).$$ Uniqueness of $\varphi$ shows that $\varphi$ is similar to ${{\left\langle{ab,-a,-b}\right\rangle}}\perp\psi$, hence it contains a subform similar 