E6, E7, E8, and the Exceptional Jordan Algebra.
The automorphism group of J is F4, which is composed
of the following bits: a triality related SO(8); the real subgroup SO(3);
and for each generator of SO(3) there are 7 addition dimensions to F4
arising from commutators of generators of SO(8) with generators of SO(3).
That yields (see page 14) 28 + 3 + 3x7 = 52 dimensions.
In complexifying J we have to replace SO(3) with SU(3). In this case
it is not too difficult to prove that only the 6 dimensions of SU(3) corresponding
to off-diagonal generators will have nonzero commutators with generators of SO(8),
and again each will give rise to 7 extra dimensions. That yields
28 + 8 + 6x7 = 78 dimensions, which makes this E6 (note: while it is
possible explicitly to construct representations using this method, I can not
convince myself that it is worth the effort for me to present them here, so
this is left as an exercise for the interested reader).
In quaternifying J we have to replace SO(3) with Sp(3).
This 21-dimensional group has 12 off-diagonal generators, and in
like fashion this will yield, when commuted with SO(8), a total of
28 + 21 + 12x7 = 133 dimensions. That's E7.
Finally, in octonifying J (which means tensoring with a new copy
of the octonions, one that commutes with the old copy), we replace SO(3)
with the 52-dimensional F4. The 28-dimensions of F4
corresponding to SO(8) are diagonally generated. The remaining 24 dimensions
are off-diagonally generated, and each of these, when commuted with
our original SO(8) gives 7 new generators. That yields 28 + 52 + 24x7 = 248
dimensions. That's E8. I will confess that I understand this
one the least.
One last conjectural note: since much of my work in this field involved
the tensor product of the complex algebra with the quaternions AND octonions
(all three, while the exceptional groups arise from tensoring pairs), I wondered
what we'd get if we tensored J (which contains the octonions) with both
the complexes and quaternions. I assume we should then replace SO(3) with SU(6),
but represented using complex quaternions. If everything works out the same,
this 35-dimensional group will also have 24 off-diagonal generators (don't forget:
this is a representation of SU(6) using 3x3 complexified and quaternionized matrices).
That would yield 28 + 35 + 24x7 = 231 dimensions. The question then is,
if this reasoning is right, what's the group? SO(22)? Is the reasoning right?
To this point I've been too lazy to find out.
Octoshop III page.