Page 14.
SO(8), Triality, F_{4}, and the Exceptional Jordan Algebra.
Let a,b,c be arbitrary real numbers, X,Y,Z arbitrary octonions. Define J
to be the set of all matrices of the form:
If A and B are elements ofJ, then so is
A o B = (AB + BA)/2.
J together with this product is generally known as the exceptional Jordan algebra
(it is the only Jordan algebra not representable by a set of real matrices).
The five exceptional Lie groups are G_{2} (14d), F_{4} (52d), E_{6} (78d),
E_{7} (133d), E_{8} (248d). They are all intimately linked to the octonions.
G_{2} is the automorphism group of the octonions, and F_{4} is the automorphism group of
J.
It is not too difficult to prove that if G maps J to J,
and
G[A^{2}] =
(G[A])^{2}
for all A in J, then G is an automorphism of J;
that is, an element of F_{4}.
The Lie group SO(3), consisting of 3x3 real orthogonal matrices, accounts for 3 of the 52 dimensions
of F_{4}. The remaining 49 dimensions all involve the octonions. As we will see, with the help
of triality the SO(8) group of actions on the octonions also appears as a subgroup of F_{4}.
This accounts for 28 more dimensions, leaving 21.
Using notation from my
book, let Tri denote the triality map, which takes SO(8) into SO(8).
Let g_{0}, g_{1} = Tri{g_{0}}, g_{2} = Tri{g_{1}}
be a triality triplet of SO(8) maps (one more application of Tri gets us back to g_{0}).
For all octonions X and Y, these maps satisfy the almost automorphic relation:
g_{0}[X]g_{1}[Y] = (g_{2}[(XY)^{*}])^{*},
which is also true for any cyclic permutation of the indices {0,1,2} of the maps g_{k}, k = 0,1,2,
since g_{0} is arbitrary. Therefore it is not too difficult to prove that the SO(8) map on
J which takes
a  Z  Y^{*} 
  
a  g_{2}[Z] 
(g_{1}[Y])^{*} 
Z^{*}  b  X 
    > 
(g_{2}[Z])^{*}  b 
g_{0}[X] 
Y  X^{*}  c 
  
g_{1}[Y]  (g_{0}[X])^{*} 
c 
is an automorphism (square it), hence an element of F_{4}.
Let U = u_{0} + u be a unit octonion, with u_{0} its real part, and u its imaginary part. Then for any octonion X, the actions
U_{L}[X] = UX,
U_{R}[X] = XU,
are examples of SO(8) rotations on the components of X. The triality map does the following:
Tri^{3}{U_{L}} = Tri^{2}{U_{R}} =
Tri{U^{*}_{L}U^{*}_{R}} = U_{L}.
Let U be the 3x3 diagonal matrix with {U^{*}, U, 1} on the diagonal, and
U^{*} its inverse, the diagonal matrix with {U, U^{*}, 1} on the diagonal.
Ok, in the triality defined SO(8) action on the Jordan matrix defined at left
(denote the initial matrix A), if we let g_{0} = U_{L}, then I'll leave it to you to prove
that the resulting action on A is the same as the action
A > UAU^{*}
(this same action altered by permuting the diagonal components of U will yield other
elements of SO(8)).
Note: in an associative world actions of the form given above on A would form a
group, but these do not. A great deal of confusion arises re the use of octonions
in the definition of Lie group and Lie algebra actions, all of which arises
because of the nonassociativity of the octonions. This confusion dissipates
once one gets used to nesting octonion actions (see
the book for a deeper discussion),
a concept that generalizes to associative algebras, but doesn't seem necessary
(and is not, but it is more unifying and universal).
Anyway, as long as one is careful with the idea of nested actions one can take
commutators of Lie algebra actions defined in terms of the octonions. In particular,
taking the commutator of the generator of the action
A > UAU^{*},
with elements of so(3) (Lie algebra of SO(3)), implies that this action on A
is also in F_{4} if U takes one of the forms
u_{0}  u  0 

u_{0}  0  u 

1  0  0 
u  u_{0}  0 
.....OR..... 
0  1  0 
.....OR..... 
0  u_{0}  u 
0  0  1 

u  0  u_{0} 

0  u  u_{0} 
These elements account for the remaining 21 dimensions of F_{4}, giving us a total
of 3 + 28 + 21 = 52. On the next page I'll outline how to get explicit representations
of E_{6}, E_{7}, and E_{8} in a very similar manner.
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Page 15.

Contents

Octoshop III page.