%% boi-8.tex (short version)
%% Chapter 8 of the Book of Involutions

\chapter{Composition and Triality} 
\label{8.chap.comp.tria}  

The main topic of this chapter is composition algebras.  Of special
interest from the algebraic group point of view are \emph{symmetric}
compositions. In dimension~$8$ there are two such types: \emph{Okubo
  algebras}, related to algebras of degree~$3$ with unitary
involutions (type $A_2$), and \emph{para-Cayley algebras} related to
Cayley algebras (type $G_2$).  The existence of these two types is due
to the existence of inequivalent outer actions of the group $\LZ/3\LZ$
on split simply connected simple groups of type~$D_4$ (``triality''
for $\Spin_8$\index{A}{triality}), 
%%\index{A}{triality for $\Spin_8$}), 
for which the fixed elements define groups of
type~$A_2$, resp.~$G_2$. Triality is defined here through an explicit
computation of the Clifford algebra of the norm of an $8$-dimensional
symmetric composition.  As a step towards exceptional simple Jordan
algebras, we introduce in the last section twisted compositions,
generalizing a construction of Springer.  The corresponding group of
automorphisms is the semidirect product $\Spin_8\rtimes S_3$.

%%  end of boi-8.tex

%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "boi"
%%% End: 
