A Lie algebra over an algebraically closed field is called exceptional if it is constructed
from one of the root systems 
, 
, 
, 
, and 
by the Chevalley construction
(Humphreys 1977). If the field has field
characteristic other than 2 or 3, than such Lie algebras are all simple
(Steinberg 1961, p. 1120).
The exceptional Lie algebras over an arbitrary field are those corresponding to the same list of root systems as well as some corresponding
to the root system of type 
and the phenomenon of triality. Algebras of this last type
occur over some fields (finite fields, number fields) and not others (the real numbers,
algebraically closed fields).
