Given an algebraic group a subgroup is said to be a Borel subgroup if it is maximal (under inclusion) among all Zariski closed connected solvable subgroups. They appear to be minimal parabolic subgroups (that is minimal among those such that is a projective variety. All the Borel subgroups are mutually conjugate and the intersection of any two contains a maximal torus in .
The main example is or where a corresponding Borel subgroup can be taken to be the sbgroup of the upper triangular matrices in and , the subgroup of the lower triangular matrices; these two subgroups are said to be mutually opposite in the sense that their intersection is precisely the maximal torus, which is in this case the subgroup of the diagonal matrices. If the quotient homogeneous spaces is called the flag variety and for a general semisimple Lie group , is called a generalized flag variety.
If the characteristic of the ground field is zero then the tangent Lie algebra of the Borel subgroup is “the” Borel subalgebra of the Lie algebra of .