Exceptional E-type root systems

Definition

The exceptional $E$ -type root systems are the three irreducible simply laced root systems

E_6,\qquad E_7,\qquad E_8.

Their ranks are $6,7,8$ , respectively. They are “exceptional” because, unlike the $A_n$ and $D_n$ systems, they do not belong to an infinite family.

Numerical data

Type	Rank	Number of roots	Dimension of the associated simple Lie algebra
$E_6$	6	72	78
$E_7$	7	126	133
$E_8$	8	240	248

For any complex semisimple Lie algebra, its dimension is its rank plus the number of roots, because every root space is one-dimensional.

Dynkin diagrams

Each $E$ -type Dynkin diagram is a tree with one trivalent vertex. The three arms, measured by the numbers of vertices beyond that trivalent vertex, have lengths

(2,2,1)\text{ for }E_6,\qquad (3,2,1)\text{ for }E_7,\qquad (4,2,1)\text{ for }E_8.

This description distinguishes them from the forked type $D_n$ diagrams, whose two short arms both have length one.

A coordinate model for E8

In $\mathbb R^8$ with orthonormal basis $e_1,\ldots,e_8$ , the $E_8$ roots are the 112 vectors

\pm e_i\pm e_j\qquad(i<j)

together with the 128 vectors

\frac12(\pm e_1\pm\cdots\pm e_8)

having an even number of minus signs. Suitable hyperplane sections give root subsystems of types $E_7$ and $E_6$ .

Under the classification theorem, these systems correspond to the exceptional complex simple Lie algebras $\mathfrak e_6,\mathfrak e_7,\mathfrak e_8$ . In particular, root subsystems inside $E_7$ organize many of the subalgebras used in constructions involving the 133-dimensional algebra $\mathfrak e_7$ .

References

Nicolas Bourbaki, Lie Groups and Lie Algebras, Chapters 4–6, Springer, 2002, Chapter VI, §4, no. 12. Publisher record.
James E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, 1972, §§11–14. Publisher record.
John C. Baez, “Three Generations in $E_7$ ,” 2026, especially §§2, 5. arXiv record.