Lie group F4 = Aut(OP2)

Indeed associativity spoils the conventional technique of using
equivalence classes of scale invariant coordinates to construct OP².
Instead one defines OP² using the exceptional Jordan algebra J(3,O),
where points of OP² consist of those matrices of J(3,O) with vanishing
Freudenthal product (A x A = 0). The Freudenthal product is defined as:

A x B = A o B - 1/2 (A tr(B) + B tr(A) ) + 1/2 ( tr(A) tr(B) - tr( A
o B)) I

where the Jordan product A o B = 1/2( AB + BA) can be written in
terms of ordinary matrix multiplication and ' I ' denotes the identity
matrix.

Normalization of matrices which satisfy A x A = 0, give primitive
idempotent matrices which satisfy:

A²=A
tr(A)=1.


Best Regards,
~M


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