Parabolic Lie algebra

In algebra, a parabolic Lie algebra formula_1 is a subalgebra of a semisimple Lie algebra formula_2 satisfying one of the following two conditions:
These conditions are equivalent over an algebraically closed field of characteristic zero, such as the complex numbers. If the field formula_8 is not algebraically closed, then the first condition is replaced by the assumption that
where formula_11 is the algebraic closure of formula_8.