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%% Chapter 4 of the Book of Involutions

\chapter{Algebras of Degree Four}    
\label{4.chap.alg.four}

Among groups of automorphisms of central simple algebras with
involution, there are certain isomorphisms, known as \emph{exceptional
  isomorphisms}, relating algebras of low degree.  (The reason why
these isomorphisms are indeed exceptional comes from the fact that in
some special low rank cases Dynkin diagrams coincide, see
\S\ref{smalldim.subsec}.) Algebras of degree~$4$ play a central r\^ole
from this viewpoint: their three types of involutions (orthogonal,
symplectic, unitary) are involved with three of the exceptional
isomorphisms, which relate them to quaternion algebras,
$5$-dimensional quadratic spaces and orthogonal involutions on
algebras of degree~$6$ respectively. A correspondence, first
considered by Albert~\cite{AlbWNCond}, between tensor products of two
quaternion algebras and quadratic forms of dimension~$6$ arises as a
special case of the last isomorphism.

The exceptional isomorphisms provide the motivation for, and can be
obtained as a consequence of, equivalences between certain categories
of algebras with involution which are investigated in the first
section.  In the second section, we focus on tensor products of two
quaternion algebras, called \emph{biquaternion algebras}, and their
Albert quadratic forms. The third section yields a quadratic form
description of the reduced Whitehead group of a biquaternion algebra,
making use of symplectic involutions.  Analogues of the reduced
Whitehead group for algebras with involution are also discussed.

\section{Exceptional Isomorphisms}
\label{exciso.sec}

The exceptional isomorphisms between groups of similitudes of central
simple algebras with involution in characteristic different from~$2$
are easily derived from Wonenburger's theorem \thmref{canhominj}, as
the following proposition shows:


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