%% boi-3.tex (short version)
%% Chapter 3 of the Book of Involutions

\chapter{Similitudes} 
\label{3.chap.simil}  

In this chapter, we investigate the automorphism groups of central
simple algebras with involution. The inner automorphisms which
preserve the involution are induced by elements which we call
\emph{similitudes}, and the automorphism group of a central simple
algebra with involution is the quotient of the group of similitudes by
the multiplicative group of the center. The various groups thus
defined are naturally endowed with a structure of linear algebraic
group; they may then be seen as twisted forms of orthogonal,
symplectic or unitary groups, depending on the type of the involution.
This point of view will be developed in Chapter~\ref{6.chap.alg.grp}.
Here, however, we content ourselves with a more elementary viewpoint,
considering the groups of rational points of the corresponding
algebraic groups.

After a first section which contains general definitions and results
valid for all types, we then focus on quadratic pairs and unitary
involutions, where additional information can be derived from the
algebra invariants defined in Chapter~\ref{2.chap.inv.inv}.  In the
orthogonal case, we also use the Clifford algebra and the Clifford
bimodule to define Clifford groups and spin groups.

\section{General Properties}
\label{genprop.sec}

To motivate our definition of similitude for an algebra with
involution, we first consider the split case, where the algebra
consists of endomorphisms of bilinear or hermitian spaces.


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