E6 (mathematics)

As well as the complex Lie group of type E6, there are five real forms of the Lie algebra, and correspondingly five real forms of the group with trivial center (all of which have an algebraic double cover, and three of which have further non-algebraic covers, giving further real forms), all of real dimension 78, as follows:

The following 72 E6 roots are derived in this manner from the split real even E8 roots. Notice the last 3 dimensions being the same as required:

1 generator which is their chirality generator, and is the sixth Cartan generator.

In addition to the 78-dimensional adjoint representation, there are two dual 27-dimensional "vector" representations.

The symmetry of the Dynkin diagram of E6 explains why many dimensions occur twice, the corresponding representations being related by the non-trivial outer automorphism; however, there are sometimes even more representations than this, such as four of dimension 351, two of which are fundamental and two of which are not.

Thus, one can get the Standard Model's elementary fermions and Higgs boson.