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 of g {\displaystyle {\mathfrak {g}}} ; the Killing perp of p {\displaystyle {\mathfrak {p}}} in g {\displaystyle {\mathfrak {g}}} is the nilradical of p {\displaystyle {\mathfrak {p}}}. Wikipedia