Parabolic Lie algebra

In algebra, a parabolic Lie algebra $$\mathfrak p$$ is a subalgebra of a semisimple Lie algebra $$\mathfrak g$$ 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 $$\mathbb F$$ is not algebraically closed, then the first condition is replaced by the assumption that where $$\overline{\mathbb F}$$ is the algebraic closure of $$\mathbb F$$.
 * $$\mathfrak p$$ contains a maximal solvable subalgebra (a Borel subalgebra) of $$\mathfrak g$$;
 * the orthogonal complement with respect to the Killing form of $$\mathfrak p$$ in $$\mathfrak g$$ is the nilradical of $$\mathfrak p$$.
 * $$\mathfrak p\otimes_{\mathbb F}\overline{\mathbb F}$$ contains a Borel subalgebra of $$ \mathfrak g\otimes_{\mathbb F}\overline{\mathbb F}$$

Examples
For the general linear Lie algebra $$\mathfrak{g}=\mathfrak{gl}_n(\mathbb F)$$, a parabolic subalgebra is the stabilizer of a partial flag of $$\mathbb F^n$$, i.e. a sequence of nested linear subspaces. For a complete flag, the stabilizer gives a Borel subalgebra. For a single linear subspace $$\mathbb F^k\subset \mathbb F^n$$, one gets a maximal parabolic subalgebra $$\mathfrak p$$, and the space of possible choices is the Grassmannian $$\mathrm{Gr}(k,n)$$.

In general, for a complex simple Lie algebra $$\mathfrak g$$, parabolic subalgebras are in bijection with subsets of simple roots, i.e. subsets of the nodes of the Dynkin diagram.