Constructions
One can construct the (compact form of the) E8 group as the automorphism group of the corresponding e8 Lie algebra. This algebra has a 120-dimensional subalgebra so(16) generated by Jij as well as 128 new generators Qa that transform as a Weyl–Majorana spinor of spin(16). These statements determine the commutators
as well as
while the remaining commutator (not anticommutator!) is defined as
It is then possible to check that the Jacobi identity is satisfied.
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