Parabolic Lie algebra
In algebra, a parabolic Lie algebra p {\displaystyle {\mathfrak {p}}} is a subalgebra of a semisimple Lie algebra g {\displaystyle {\mathfrak {g}}} satisfying one of the following two conditions: p {\displaystyle {\mathfrak {p}}} contains a maximal solvable subalgebra (a Borel subalgebra) of g {\displaystyle {\mathfrak {g}}} ; the orthogonal complement with respect to the Killing form of p {\displaystyle {\mathfrak {p}}} in g {\displaystyle {\mathfrak {g}}} is isomorphic to the nilradical of p {\displaystyle {\mathfrak {p}}} . These conditions are equivalent over an algebraically closed field of characteristic zero, such as the complex numbers.