central simple algebras of degree divisible by $4$ in the following sections.

Algebras of degree 4 {#deg4.sec}
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Throughout this section, $A$ is a central simple algebra of degree $4$ over an arbitrary field $F$ of characteristic different from $2$. Involutions on $A$ are classified by cohomological invariants (see [@KMRT Ch. 4]), which we use to give an explicit description of the involutions on $A$ that become hyperbolic over $F_Q$.

The orthogonal case {#deg4orth.sec}
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This case is easy to handle using Clifford algebras.

\[deg4orth.prop\] Let $\sigma$ be an orthogonal involution on $A$.

1.  $(A,\sigma)$ is hyperbolic if and only if $\operatorname{disc}\sigma=1$ and one of the components of $C(A,\sigma)$ is split;

2.  $(A,\sigma)$ contains ${(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}$ if and only if $\operatorname{disc}\sigma=1$ and one of the components of $C(A,\sigma)$ is isomorphic to $Q$;

3.  $A$ contains $Q$ if and only if $A\otimes_FQ$ is Brauer-equivalent to a quaternion algebra.

\(1) This readily follows from [@BST 2.5].

\(2) If $(A,\sigma)$ contains ${(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}$, then there is a quaternion $F$-algebra $Q'$ with canonical involution ${\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}}$ such that $$(A,\sigma)\simeq{(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}\otimes(Q',{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}}).$$ By [@KMRT (15.12)], this relation holds if and only if $C(A,\sigma)\simeq Q\times Q'$. This proves (2).

\(3) If $A$ contains (a copy of) $Q$, then the centralizer of 