%% boi-10.tex (short version)
%% Chapter 10 of the Book of Involutions

\chapter{Trialitarian Central Simple Algebras}  
\label{10.chap.tria.alg}   

We assume in this chapter that $F$ is a field of characteristic
not~$2$.  Triality for $\gPGO^+_8$\index{A}{triality},
%%\index{A}{triality for $\gPGO^+_8$},
i.e., the outer action of~$S_3$ on~$\gPGO^+_8$ and its consequences,
is the subject of this chapter.  In the first section we describe the
induced action on~$H^1(F, \gPGO^+_8)$.  This cohomology set classifies
ordered triples $(A,B,C)$ of central simple algebras of degree~$8$
with involutions of orthogonal type such that
$\bigl(C(A,\sigma_A),\underline{\sigma}\bigr) \isom (B,\sigma_B)
\times (C,\sigma_C)$.  Triality implies that this property is
symmetric in $A$, $B$ and $C$, and the induced action of~$S_3$
on~$H^1(F, \gPGO^+_8)$ permutes $A$, $B$, and~$C$.  As an application
we give a criterion for an orthogonal involution on an algebra of
degree~$8$ to decompose as a tensor product of three involutions.

We may view a triple $(A,B,C)$ as above as an algebra over the split
\'etale algebra $F\times F\times F$ with orthogonal involution
$(\sigma_A,\sigma_B,\sigma_C)$ and some additional structure (the fact
that $\bigl(C(A,\sigma_A),\underline{\sigma}\bigr) \isom (B,\sigma_B)
\times (C,\sigma_C)$). Forms of such ``algebras,'' called trialitarian
algebras, are classified by $H^1(F, \gPGO^+_8\rtimes S_3 )$.
Trialitarian algebras are central simple algebras with orthogonal
involution of degree~$8$ over cubic \'etale $F$-algebras with a
condition relating the central simple algebra and its Clifford
algebra. Connected components of automorphism groups of such
trialitarian algebras give the outer forms of simple adjoint groups of
type~$D_4$.

Trialitarian algebras also occur in the construction of Lie algebras
of type~$D_4$. Some indications in this direction are given in the last
section.


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