Simply laced root system

Definition

A reduced root system $\Phi$ is simply laced if every edge of its Dynkin diagram is a single unoriented edge. Equivalently, the roots in each irreducible component all have the same length. Different orthogonal components may be rescaled independently, so roots in distinct components need not have equal lengths.

Classification

The irreducible simply laced root systems are precisely

A_n\ (n\geq 1),\qquad D_n\ (n\geq4),\qquad E_6,E_7,E_8.

Thus “ADE type” and “simply laced type” are synonymous for reduced finite root systems. A reducible root system is simply laced exactly when each irreducible component is of one of these types.

Root geometry

Normalize an irreducible simply laced system so that $\langle\alpha,\alpha\rangle=2$ for every root. For distinct nonopposite roots,

\langle\alpha,\beta\rangle\in\{-1,0,1\}.

In particular, orthogonal roots have neither their sum nor their difference as a root. This makes orthogonality especially effective in calculations with root subsystems and centralizers.

The classical simply laced families are the type $A$ and type $D$ systems; the remaining cases are the exceptional $E$ -type systems.

References

James E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, 1972, §§9–12. Publisher record.
Nicolas Bourbaki, Lie Groups and Lie Algebras, Chapters 4–6, Springer, 2002, Chapter VI, §§1, 4. Publisher record.
John C. Baez, “Three Generations in $E_7$ ,” 2026, §§3, 5. arXiv record.