There are five real forms of the exceptional Lie group, $E\_6$. Four of them are notated as in the following:


* The split form as EI or $E\_{6(6)}$
* The quasi-split form as EII or $E\_{6(2)}$
* EIII or $E\_{6(-14)}$
* EIV or $E\_{6(-26)}$


What do the annotations to $E\_6$ actually indicate and are they also used for real forms for Lie groups in general?



What is meant by this notation of the real forms of $E_6$?
The notation is a bit complicated to make precise, but the number in parentheses is the *character*, which is defined on page 353 section C of Helgason, **Differential Geometry, Lie Groups and Symmetric Spaces**. The character is the difference $\dim \mathfrak{p}\_0 - \dim \mathfrak{k}\_0$ in dimensions in a Cartan decomposition of a real form. As Helgason explains, the character of an exceptional real simple Lie algebra determines the Lie algebra, but for the classical real simple Lie algebras, it doesn't, so the notation is only used for exceptionals.


