$Q$. In particular, $A$ is not division, by Proposition \[division.prop\], so $A\simeq M_2(Q')$ for some quaternion $F$-algebra $Q'$, and $Q\otimes_FQ'$ is a division algebra by Proposition \[deg4orth.prop\](3). As observed in the proof of Proposition \[split.prop\](2), $(A,\sigma)$ contains $(Q',{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})$, hence $$(A,\sigma)\simeq\operatorname{Ad}_{{{\langle\!\langle{\lambda}\rangle\!\rangle}}}\otimes(Q',{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})
  \qquad\text{for some $\lambda\in F^\times$}.$$ The algebra $A$ carries a hyperbolic symplectic involution $\tau$, and we have by [@KMRT (16.21)] $$\Delta_\tau(\sigma)=(\lambda)\cup [Q'].$$ Since $(A,\sigma)_{F_Q}$ is hyperbolic, this invariant vanishes over $F_Q$. Hence, by [@Lam X(4.11)] and Arason’s common slot lemma we may assume as in the proof of Lemma \[deg4symp.lem\] that $(\lambda)\cup
  [Q']=(\lambda)\cup [Q]$. Therefore, ${{\langle\!\langle{\lambda}\rangle\!\rangle}}\cdot
  n_{Q'}\simeq{{\langle\!\langle{\lambda}\rangle\!\rangle}}\cdot n_Q$, which shows that $(A,\sigma)$ is as in case (b).

It is clear that $(A,\sigma)_{F_Q}$ is hyperbolic in case (a). In case (b) the algebra $A$ carries a hyperbolic symplectic involution $\tau$ and $\Delta_\tau(\sigma)=(\lambda)\cup[Q']$ vanishes over $F_Q$. Therefore, $(A,\sigma)_{F_Q}$ is hyperbolic.

$F_Q$-minimal quadratic forms of dimension 5 {#min.sec}
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A quadratic form $\varphi$ over $F$ is called *$F_Q$-minimal* if $\varphi_{F_Q}$ is isotropic and $\psi_{F_Q}$ is anisotropic for every proper subform $\psi\subset\varphi$. In this section, we show that Theorem \[deg4symp.thm\] can be used to recover (and is in fact equivalent to) the description of $F_Q$-minimal forms of dimension $5$ due to Hoffmann, Lewis, and Van 