Let , the group of all automorphisms of the -dimensional vector space over the field of complex numbers , and a subgroup of . The standard Borel subgroup of is the subgroup of consisting of all upper triangular matrices (in ). A Borel subgroup of is a conjugate (in ) of the standard Borel subgroup of .

The notion of a Borel subgroup can be generalized. Let be a complex semi-simple Lie group. Then any maximal solvable subgroup is called a Borel subgroup. All Borel subgroups of a given group are conjugate. Any Borel group is connected and equal to its own normalizer, and contains a unique Cartan subgroup. The intersection of with a maximal compact subgroup of is the maximal torus of .