 For a linear algebraic group formula_21 its Borel subgroup is defined as a subgroup which is closed, connected, and solvable in formula_21, and it is the maximal possible subgroup with these properties (note the second two are topological properties).For example, in formula_55 and formula_56 the group of upper-triangular, or lower-triangular matrices are two of the Borel subgroups.The example given above, the subgroup formula_57 in formula_58 is the Borel subgroup.