{{\langle\!\langle{\mu,a,b'}\rangle\!\rangle}}={{\left\langle{1,-\mu,-a,\mu b'}\right\rangle}}\perp{{\left\langle{\mu
  a,ab',-\mu ab'}\right\rangle}}\perp{{\left\langle{-b'}\right\rangle}}$$ is hyperbolic. On the other hand, by , the quadratic form ${{\left\langle{1,-b',-aa',-\mu}\right\rangle}}$ is isotropic. Multiplying by $(-ab'\mu)$, we get that $-a'b'\mu$ is represented by the form ${{\left\langle{\mu a, ab',-\mu ab'}\right\rangle}}$. Hence, in view of , the quadratic form ${{\left\langle{1,-\mu,-a,\mu b',-a'b'\mu}\right\rangle}}$, which is a $5$-dimensional subform of ${{\langle\!\langle{\mu,a,b'}\rangle\!\rangle}}$, is necessarily isotropic. Since the quadratic forms ${{\left\langle{1,-a}\right\rangle}}$ and ${{\left\langle{1,-b',a'b'}\right\rangle}}$ are anisotropic, this completes the proof.

Using Proposition \[deg4symp.prop\], we may characterize the symplectic involutions on $A$ that become hyperbolic over $F_Q$:

\[deg4symp.thm\] Let $\sigma$ be a symplectic involution on $A$. If $(A,\sigma)_{F_Q}$ is hyperbolic, then either

1.  $(A,\sigma)\simeq{(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}\otimes(Q',\rho)$ for some quaternion $F$-algebra with orthogonal involution $(Q',\rho)$, or

2.  $(A,\sigma)\simeq\operatorname{Ad}_{{{\langle\!\langle{\lambda}\rangle\!\rangle}}}\otimes(Q',{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})$ for some quaternion $F$-algebra $Q'$ and some $\lambda\in F^\times$ with the following properties: $Q\otimes_FQ'$ is a division algebra and the norm forms $n_Q$, $n_{Q'}$ satisfy ${{\langle\!\langle{\lambda}\rangle\!\rangle}}\cdot n_Q\simeq{{\langle\!\langle{\lambda}\rangle\!\rangle}}\cdot n_{Q'}$.

Conversely, if $(A,\sigma)$ is of either type above, then $(A,\sigma)_{F_Q}$ is hyperbolic.

Note that (a) and (b) are mutually exclusive since in the first case $A\otimes_FQ$ has Schur index $1$ or $2$, whereas it has index $4$ in case (b).

Suppose $(A,\sigma)$ is not as in case (a) and $(A,\sigma)_{F_Q}$ is hyperbolic. Then Proposition \[deg4symp.prop\] shows that $A$ does not contain 