More generally, the stabilizer of a flag (the linear operators on "V" such that formula_5 for all "i"&hairsp;) is, in matrix terms, the algebra of block upper triangular matrices (with respect to an adapted basis), where the block sizes formula_6. The stabilizer subgroup of a complete flag is the set of invertible upper triangular matrices with respect to any basis adapted to the flag. The subgroup of lower triangular matrices with respect to such a basis depends on that basis, and can therefore "not" be characterized in terms of the flag only.